MGWS (ô $(,048<@DHLPTX\`dhlptx|€„ˆŒ”˜œ ¤¨¬°´¸¼ÀÄÈÌÐðLXT`Xh|€„ø(,04,ùäù,ú@ùùˆü¸ø°øÈø4ùœøÀø@û¼yŒùØø|„ùdøÔøÜø€ø¨øˆø|ø„ø˜øüùèùÈ0d˜œ ¨¬¤°´¸ð:9ôütxaàl#¨)˜)p)(-
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	__sig__10`ˆû)§è§ûÿÿÿœü)"kThe automorphism (x,y,...) -> (x+p,y,...) where p is a polynomial on S or its ambient space not involving xLdý)KßXûÿûÿûÿûÿü§§ð§xÿ)§ø§ûÿÿÿÄ*FThe projectivity of A determined by the matrix of base ring elements M *ïÿëYûÿûÿûÿûÿ¨§¨@*
	__sig__12@`*§¨ûÿÿÿÀ*
Projectivityà*ÿÿÿÿÿÿÿÿ¨*"The Nagata automorphism of 3-space!0!*ë0ûÿûÿûÿûÿ$¨§¨%*
	__sig__13ì&'	*§ ¨ûÿÿÿ)
*NagataAutomorphism,0,*ÿÿÿÿÿÿÿÿ¨*:A linear endomorphism l and a translation t with f = t o l606*000ûÿûÿûÿ<¨§0¨ :*
	__sig__14¨;Ð;*§8¨ûÿÿÿð<0*
__sig__0ierstrass*HThe birational pullback of X under the standard quadratic transformationBhB¨B*ßßûÿûÿûÿûÿT¨L¨§@¨(G*
	__sig__150IxI*§H¨ûÿÿÿôK* cThe birational pullback of X quadratic transformation of P in the linearly inde pendent points of SQ4Q*ÿíßßûÿûÿûÿûÿ\¨§P¨€T*
	__sig__16@V`V*§X¨ûÿÿÿ˜X*4/usr/local/Magma2.8/package/AlgGeom/map/curve_maps.m	ð	*4/usr/local/Magma2.8/package/AlgGeom/map/curve_maps.m	ð	* aTranslate the point p to the point [0:1:0] and the tangent line to C at p to the line at infinityDd´d*íéé0ûÿûÿûÿt¨d¨h¨Œg*
__sig__0jk<k*d¨p¨ûÿÿÿœl*TranslationToInfinityÜm*ÿÿÿÿÿÿÿÿl¨ *3True iff C is an elliptic curve in Weierstrass formu¨u!*éûÿûÿûÿûÿŒ¨d¨€¨øx"*
__sig__1z@zXz#*d¨ˆ¨ûÿÿÿÄ|$*IsEllipticWeierstrass%*ÿÿÿÿÿÿÿÿ„¨&*7True iff C is a hyperelliptic curve in Weierstrass formˆ<ˆ¤ˆ'*éûÿûÿûÿûÿ¤¨d¨˜¨Ô(*
__sig__2àø)*d¨ ¨ûÿÿÿœ’**IsHyperellipticWeierstrass”<”+*ÿÿÿÿÿÿÿÿœ¨,*9/usr/local/Magma2.8/package/AlgGeom/linearsystem/linsys.mà	ð	-*9/usr/local/Magma2.8/package/AlgGeom/linearsystem/linsys.m8	ð	.*The base scheme of L¡Ô¡1*´¨À¨ûÿÿÿT¤)ßâÿTûÿûÿûÿûÿl¤\¤`¤À
)
__sig__0ÀØ)\¤h¤ûÿÿÿd$)&xAn element of the function field of the ambient space containing the curve C corresponding to f which agrees with f on C¬Ä&)
__sig__2¨À')\¤˜¤ûÿÿÿà()
FFCurveToPlane))ÿÿÿÿÿÿÿÿ”¤*)VAn element of the function field of C corresponding to the ambient rational function f +)TéÕÿûÿûÿûÿûÿ´¤\¤¨¤Ü,)
__sig__3 Ô !-)\¤°¤ûÿÿÿœ".)
FFPlaneToCurve<$/)ÿÿÿÿÿÿÿÿ¬¤0)0¢True if there is an element of the function field of the ambient of C corresponding to the quotient f of two ambient polynomials, together with that element if so 1H11)þÿßTûÿûÿûÿÌ¤\¤À¤p82)
__sig__49ø9:3)\¤È¤ûÿÿÿ€;4)
IsAmbientFFElt<5)ÿÿÿÿÿÿÿÿÄ¤6)"hTrue iff there is an element of the function field C corresponding to f together with that element if soDEE7)þÿéáûÿûÿûÿä¤\¤Ø¤xI8)
__sig__5JJK9)\¤à¤ûÿÿÿÄM:)
IsCurveFFEltO\O;)ÿÿÿÿÿÿÿÿÜ¤<)$oAn element of the function field of the ambient of C corresponding to the quotient f of two ambient polynomials˜VÐVW=)þÿßTûÿûÿûÿûÿü¤\¤ð¤Ð[>)
__sig__6]L]t]?)\¤ø¤ûÿÿÿL`@)
AmbientFFElta\aA)ÿÿÿÿÿÿÿÿô¤B)8ÁAn element of the function field of the ambient of C corresponding to f (which can be a curve function field element, an ambient function field element or a quotient of two ambient polynomials)´lÌlC)þÿéáûÿûÿûÿ¥\¤¥ÈrD)
__sig__7t€t tE)\¤¥ûÿÿÿØuF)

CurveFFEltôvwG)ÿÿÿÿÿÿÿÿ¥H)$qTrue iff f is in the coordinate ring of the ambient space of S, in which case return the corresponding polynomial`xI)þÿßKûÿûÿûÿ,¥\¤ ¥Ä†J)
__sig__8ˆ¼ˆÔˆK)\¤(¥ûÿÿÿDL)IsAmbientFunction¼ÔM)ÿÿÿÿÿÿÿÿ$¥X)
__sig__0FunctionN)=True iff f in is the function field (or coordinate ring) of A—O)þÿtûÿûÿûÿûÿD¥\¤8¥dœP)
__sig__9ÄäQ)\¤@¥ûÿÿÿ R)IsAmbientRationalFunction$¢<¢Û(>ÛA is an ambient affine or projective scheme matching that of S but possibly having a different base ring. Return the scheme defined by the equations of S in A subject to coercion of coefficients or the coefficient map m¬Äë(ýÿ}ûÿûÿûÿ|£´£d£¨£ôì(
__sig__6Øðí(d£°£ûÿÿÿ ß(}ûÿûÿûÿ|Æ„£d£x£@è(ßßßûÿûÿûÿÄ£¼£˜£¤£ ï(˜£¸£ûÿÿÿ„ð(FThe product of projective spaces whose dimensions are the entries of N!ñ(ýÿßûÿûÿûÿÌ£˜£À£¼$ò(
__sig__1&Œ&Ô&ó(˜£È£ûÿÿÿ¨(ü(
__sig__3ity+¸+ô(HThe hyperplane at infinity as a subscheme of the projective closure of X0à01õ(ßßûÿûÿûÿûÿÜ£˜£Ð£À7ö(
__sig__29È9à9÷(˜£Ø£ûÿÿÿ0;ø(HyperplaneAtInfinity<<ù(ÿÿÿÿÿÿÿÿÔ£ú(>The line at infinity as a curve in the projective closure of ABû(ëéûÿûÿûÿûÿô£˜£è£˜Fý(˜£ð£ûÿÿÿxIþ(
LineAtInfinityKÿ(ÿÿÿÿÿÿÿÿì£)QA standard affine patch of P containing p together with the point p in that chart`RxR)íßßíûÿûÿûÿ¤˜£¤`V)
__sig__4WðWX)˜£¤ûÿÿÿ Z)
__sig__6ches\)LThe number of standard affine patches of S, returning 0 for an affine scheme`ì`)ßûÿûÿûÿûÿ¤˜£¤´d)
__sig__5eœe´e)˜£¤ûÿÿÿÔf)NumberOfAffinePatchesœj	)ÿÿÿÿÿÿÿÿ¤
) _A sequence of maps from affine spaces whose images partition P together with the returned pointhss¨s)êíûÿûÿûÿ4¨4¤˜£(¤w)˜£0¤ûÿÿÿ¨x)AffineDecomposition@zXz)ÿÿÿÿÿÿÿÿ,¤)DAn affine patch of C centred at the point p and the embedding into C‚ˆ‚)íßß0ûÿûÿûÿL¤˜£@¤$‰)
__sig__7,D)˜£H¤ûÿÿÿÔ)CentredAffinePatch‘h‘)ÿÿÿÿÿÿÿÿD¤)3/usr/local/Magma2.8/package/AlgGeom/scheme/ff_amb.m@)	*	)3/usr/local/Magma2.8/package/AlgGeom/scheme/ff_amb.m)	*	),•Return the numerator and denominator of the ambient rational function f made homogeneous as elements of the coordinate ring of the ambient space of X|¢”(Minus†•(ÿÿÿÿÿÿÿÿD¢–(
$ŒhŒ—(íûÿûÿûÿûÿlÆd¢¢X¢ø˜(
__sig__3’0’H’™(¢`¢ûÿÿÿÌ“¬(
aresesš("gThe nonzero coordinates of p as a sequence together with a sequence of the indices of these coordinatesœ<œdœ›(íûÿûÿûÿt¢¢h¢œ k*ÿíêéûÿûÿûÿûÿ´©l©¨©Hl*
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m*l©°©ûÿÿÿ˜n*HThe unique hypersurface of degree d of S passing through the points of P`o*ÿíßûÿûÿûÿûÿÄ©l©¸©\p*
__sig__44Lq*l©À©ûÿÿÿ„r*
SchemeThroughœs*ÿÿÿÿÿÿÿÿ¼©t*'The line through p and q as points of C|¤¼u*ííééûÿûÿûÿûÿÜ©l©Ð©<v*
__sig__5 h w*l©Ø©ûÿÿÿ"x*?/usr/local/Magma2.8/package/AlgGeom/linearsystem/image_linsys.m¸;	y*?/usr/local/Magma2.8/package/AlgGeom/linearsystem/image_linsys.m¸;	z*WThe scheme defined by polynomials of degree d in the codomain of f which vanish on f(X)l0œ0´0{*ßûÿûÿûÿûÿô©ä©è©Œ7|*
__sig__09d9œ9}*ä©ð©ûÿÿÿü:~*VThe linear system of hypersurfaces of degree d in the codomain of f which contain f(X)l>*ßûÿûÿûÿûÿªä©ø©¤B€*
__sig__1DüDE*ä©ªûÿÿÿDF‚*
ImageSystemdH¤Hƒ*ÿÿÿÿÿÿÿÿü©„*,/usr/local/Magma2.8/package/AlgGeom/k3/AFR.m3	¸;	…*,/usr/local/Magma2.8/package/AlgGeom/k3/AFR.m4	¸;	‡*/The indices of singularities in the basket of Xbe their ind†*bûÿûÿûÿûÿ ªÌ«ª‰*
	__sig__45ç__siˆ*Ì«$ªûÿÿÿûÿÿÿŠ*
SetIdentifiersø_‹*ÿÿÿÿÿÿÿÿ¯Œ*SSet the numbers used in the lists of Altinok-Fletcher-Reid to the K3 surfaces in DB˜e°e*ûÿûÿûÿûÿûÿ<ªª0ª8kŽ*
__sig__1l0lHl*ª8ªûÿÿÿ¨m‘*ÿÿÿÿÿÿÿÿ4ª’*0/usr/local/Magma2.8/package/AlgGeom/k3/baskets.mu@7	¸;	“*0/usr/local/Magma2.8/package/AlgGeom/k3/baskets.mw7	¸;	”*zÇThe first return sequence contains baskets of singularities for virtual K3 surfaces polarised by a divisor with a g-dimensional linear system (or bigger) and at most n curves in the resolution of its singularities where g is the corresponding entry of the second sequence. The third sequence is the corresponding degree, while (if Proof is true) the fourth contains elements which had positive degree but negative coefficients for the indicated value of g”‘´‘Ì‘•*ûÿ\ªLªPª°•–*
__sig__0˜X˜(™—*LªXªûÿÿÿðš™*ÿÿÿÿÿÿÿÿTªš*0/usr/local/Magma2.8/package/AlgGeom/k3/centres.m È:	¸;	›*0/usr/local/Magma2.8/package/AlgGeom/k3/centres.m¤;	¸;	ž*
__sig__0¥0¥H¥ *
Centres ¦È¦à¦º)5/usr/local/Magma2.8/package/AlgGeom/map/map_closure.m	8J	»)5/usr/local/Magma2.8/package/AlgGeom/map/map_closure.m	8J	¼)<The restriction of f to the jth affine patch of its codomain½)00ûÿûÿûÿûÿ§ü¦ì¦ð¦¼¿)ì¦ø¦ûÿÿÿÜÊ)
__sig__0äüË)§(§ûÿÿÿ|Ì)$The constant map A -> B with image p\Í)íßß0ûÿûÿûÿûÿ<§§0§LÎ)
__sig__1¤¼Ï)§8§ûÿÿÿh Ð)
ConstantMap°!è!Ñ)ÿÿÿÿÿÿÿÿ4§Ò)HThe restriction of f to the scheme X in its domain and to new codomain Y(Œ(¤(Ó)ßß00ûÿûÿûÿûÿT§§H§¼.Ô)
__sig__20Ü01Õ)§P§ûÿÿÿÌ4ä)
__sig__5rphism,7Ö)The identity automorphism on XÜ9×)ßXûÿûÿûÿûÿd§§X§¼<Ø)
__sig__3=ô=>Ù)§`§ûÿÿÿ¤?Ú)IdentityAutomorphismA¼AÛ)ÿÿÿÿÿÿÿÿ\§Ü)BThe automorphism of S that permutes its coordinates according to g,ItIÝ)åßXûÿûÿûÿûÿ|§§p§XNÞ)
__sig__4OÐOèOß)§x§ûÿÿÿ Qà)PermutationAutomorphismSá)ÿÿÿÿÿÿÿÿt§â)BThe automorphism of S that permutes its coordinates according to gÔYZã)åßXûÿûÿûÿûÿœ§”§§ˆ§8_å)§§ûÿÿÿè`æ):The automorphism determined by the sequence F of functionsðbxcç)ß0ûÿûÿûÿûÿ¬§¤§§˜§Ðfè)
__sig__6gh(hé)§ §ûÿÿÿlê)FThe automorphism of P determined by the matrix of base ring elements M¼oë)ïÿßXûÿûÿûÿûÿô§´§§¨§tuì)
__sig__7v\v€ví)§°§ûÿÿÿxî)(The matrix determining f if it is linearzlz”zï)Yïÿûÿûÿûÿûÿ¸èÄ§§¸§tð)
__sig__8€4„ñ)§À§ûÿÿÿ`†þ)
	__sig__11ationˆò)KThe quadratic automorphism [x,y,...] -> [1/x,1/y,...] of projective space P$Xó)êXûÿûÿûÿûÿä§Ô§§È§È’ô)
__sig__9“à“ø“õ)§Ð§ûÿÿÿ@–ö)QuadraticTransformationš÷)ÿÿÿÿÿÿÿÿÌ§ø)IThe quadratic transformation of P in the linearly independent points of S€ ˜ ù)ÿíêXûÿûÿûÿûÿD¨ì§§à§è¤2*
mposition…)¼¥¦ûÿÿÿ¬)__sig__0singularPointd†)4°The image of the projection of X away from the nonsingular point p of X together with the locus in the image which is the blowup of p on X (returned as a point if X is a curve)x‡)íßß0ßûÿûÿ$¦¼¥¦ˆ)
__sig__5$<‰)¼¥ ¦ûÿÿÿ¼Š)ProjectionFromNonsingularPoint\‹)ÿÿÿÿÿÿÿÿ¦Œ)1/usr/local/Magma2.8/package/AlgGeom/map/amb_map.m L	ˆX	)1/usr/local/Magma2.8/package/AlgGeom/map/amb_map.mðL	ˆX	)0ûÿûÿûÿûÿT¦D¦4¦8¦@%‘)4¦@¦ûÿÿÿ(’)

BasePoints\*t*“)ÿÿÿÿÿÿÿÿ<¦”),•The set of base points of f defined over the extension L of the base field (or the base field itself if L is omitted) if that set of points is finitel8•)õÿ0ûÿûÿûÿûÿÌ¨\¦4¦P¦ü;–)
__sig__1<Ô<ì<—)4¦X¦ûÿÿÿ¤>D*IsFreech˜)DTrue iff f defined by polynomials on the ambient space of its domainDÄD™)0ûÿûÿûÿûÿl¦4¦`¦ôHš)
__sig__2J\JtJ›)4¦h¦ûÿÿÿˆMœ)
IsPolynomialNøN)ÿÿÿÿÿÿÿÿd¦ž),True iff the domain and range of f are equalSŒSŸ)0ûÿûÿûÿûÿ„¦4¦x¦X )
__sig__3YÔYZ¡)4¦€¦ûÿÿÿ\¢)
IsEndomorphismÐ]£)ÿÿÿÿÿÿÿÿ|¦¤)ZThe degree of the functions which define f assuming its domain and codomain are projectiveøde¥)0ûÿûÿûÿûÿœ¦4¦¦øg¦)
__sig__4k k¸k§)4¦˜¦ûÿÿÿm¨)
FunctionDegreen©)ÿÿÿÿÿÿÿÿ”¦ª)6True iff f is a morphism defined by linear polynomialsÔu«)0ûÿûÿûÿûÿ´¦4¦¨¦Hy¬)
__sig__5zlz”z­)4¦°¦ûÿÿÿd}®):True iff f is defined by polynomials of degree at most one¬€Ä€¯)0ûÿûÿûÿûÿÄ¦4¦¸¦ ˆ°)
__sig__6‹ ŒdŒ±)4¦À¦ûÿÿÿ²)
IsAffineLinearŒ³)ÿÿÿÿÿÿÿÿ¼¦´)True iff f is dominantø“µ)0ûÿûÿûÿûÿÜ¦4¦Ð¦€š¶)
__sig__7›¸›Ð›·)4¦Ø¦ûÿÿÿ¨¸)

IsDominantÐž(Ÿ¹)ÿÿÿÿÿÿÿÿÔ¦¾)
__sig__0£ø£ ¤)
GenericToFOF†À†)ÿÿÿÿÿÿÿÿd¤)^Return the element of the function field of P corresponding to the generic rational function fd‘)ßâÿTûÿûÿûÿûÿ„¤\¤x¤€” )
__sig__1–X–—!)\¤€¤ûÿÿÿ€š")
GenericToFF¸›Ð›#)ÿÿÿÿÿÿÿÿ|¤%)ÕÿTûÿûÿûÿûÿœ¤\¤¤x¢¢¨¢Ø*,/usr/local/Magma2.8/package/AlgGeom/k3/sum.m[	j	Ù*,/usr/local/Magma2.8/package/AlgGeom/k3/sum.m[	j	Ú* cThe Hilbert series corresponding to a sequence of polynomials F and a sequence of starting values VxÛ*Sûÿûÿûÿûÿ@¯t«d«h«Àæ*
__sig__2ipliedByMinÞ*HGiven a sequence of polynomials determining a Hilbert polynomial with periodic corrections, together with a sequence of alternative early values, return the Hilbert series' numerator, together with a sequence containing the factors of the minimal denominator4Lß*ûÿûÿûÿ„«d«x«ü à*
__sig__1"0"h"á*d«€«ûÿÿÿ8$â*+HilbertSeriesMultipliedByMinimalDenominatorX'x'ã*ÿÿÿÿÿÿÿÿ|«ä*"lThe Hilbert function corresponding to a sequence of univariate polynomials and a sequence of starting values1¤1å*Xûÿûÿûÿûÿœ«d««9ç*d«˜«ûÿÿÿÌ:è*
HilbertFunctionü;é*ÿÿÿÿÿÿÿÿ”«ê*,/usr/local/Magma2.8/package/AlgGeom/k3/two.m`	j	ë*,/usr/local/Magma2.8/package/AlgGeom/k3/two.m`	j	ì*(†Adds additional weights to X to account for the polarisation of the singularities disregarding results of codimension bigger than cmaxKí*‹‹ûÿûÿûÿûÿ¼«¬«°«èOî*
__sig__0QHQ`Qï*¬«¸«ûÿÿÿìRð*ForceSingularitiesTT|Tñ*ÿÿÿÿÿÿÿÿ´«ò*-/usr/local/Magma2.8/package/AlgGeom/k3/type.m	j	ó*h-/usr/local/Magma2.8/package/AlgGeom/k3/type.m	j	ö*genusø_ü*steps"÷*
	numeratorÐaèa&+Ì«”¬ûÿÿÿxc;+
	__sig__10et@e'+1The elements of the K3 database DB with weights WPgˆg(+ûÿûÿûÿûÿ¨¬Ì«œ¬m)+
__sig__7mnn*+Ì«¤¬ûÿÿÿäp++K3SurfacesFromWeightsœt,+ÿÿÿÿÿÿÿÿ ¬-+/The unique K3 in the database DB with weights W¤xÌxôx.+bûÿûÿûÿûÿÀ¬Ì«´¬¤}/+
__sig__840+Ì«¼¬ûÿÿÿ„1+K3SurfaceFromWeights†`†2+ÿÿÿÿÿÿÿÿ¸¬3+<The elements of the K3 database DB with singularity basket BÐ4+ûÿûÿûÿûÿØ¬Ì«Ì¬ø’5+
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3*ûÿûÿûÿûÿÜ¨Ô¨´¨È¨4*
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__sig__2d|9*´¨à¨ûÿÿÿÜ:*?True iff the hypersurfaces of L have no common geometric points@è;*ûÿûÿûÿûÿô¨´¨è¨$<*
__sig__3%(%@%=*´¨ð¨ûÿÿÿ(F*,The hypersurface common to all elements of L+ü+G*ßûÿûÿûÿûÿ$©´¨©D1H*
__sig__534Ì4I*´¨ ©ûÿÿÿŒ7Ý,The number of vertices of GÞ,lûÿûÿûÿûÿ¸€³¸²t³@à,¸²|³ûÿÿÿX‚-indextiesá,2The sequence of selfintersections of vertices of Gâ,lûÿûÿûÿûÿ¾³¸²„³Äã,
__sig__7Øä,¸²Œ³ûÿÿÿðå,Selfintersections æ,ÿÿÿÿÿÿÿÿˆ³ç,3The multiplicities of the pullback of the base germèè,lûÿûÿûÿûÿÀ¨³¸²œ³¸	é,
__sig__8	Ì	ê,¸²¤³ûÿÿÿä	ë,
Multiplicitiesø	ì,ÿÿÿÿÿÿÿÿ ³í,JThe multiplicities of the canonical class of the germ of a surface along GL
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__sig__9!„!ð,¸²¼³ûÿÿÿÜ$ñ,
CanonicalClass`5ò,ÿÿÿÿÿÿÿÿ¸³ó,0The axis multiplicities of the nearby germs to GDŒDô,lûÿûÿûÿûÿØ³¸²Ì³°Kõ,
	__sig__10Rö,¸²Ô³ûÿÿÿäY÷,AxisMultiplicitiesˆ^ø,ÿÿÿÿÿÿÿÿÐ³-
	__sig__12es¼^ù,2The galois multiplicities of the nearby germs to Gä^ú,lûÿûÿûÿûÿ ¸ð³¸²ä³(gû,
	__sig__11<gü,¸²ì³ûÿÿÿ¸vý,GaloisMultiplicitiesœƒþ,ÿÿÿÿÿÿÿÿè³ÿ,&The germ used to create G, if assignedTŠ-líûÿûÿûÿûÿ´¸²ü³ð«-¸²´ûÿÿÿ¬-vertexraph+ûÿûÿûÿûÿP¬H¬Ì«<¬Èl+
__sig__2mÀmØm+Ì«D¬ûÿÿÿ„p+BThe database of numerical K3 surfaces having codimension at most nŒu¤u+ûÿûÿûÿûÿX¬Ì«L¬ôx+
__sig__3z<zTz+Ì«T¬ûÿÿÿÀ|+Print X at level lÌ~+bûÿûÿûÿûÿûÿh¬Ì«\¬`†+
__sig__4‡ø‡ˆ+Ì«d¬ûÿÿÿdŒ+
PrintVSrfK3„Ž¼Ž+ÿÿÿÿÿÿÿÿ`¬+*Numerical K3 corresponding to the record X“(“+‹bûÿûÿûÿûÿ¬€¬Ì«t¬ˆ™+
__sig__5šØšðš +Ì«|¬ûÿÿÿ°œ!+
	K3Surfaceø0ž"+ÿÿÿÿÿÿÿÿx¬#+2Numerical K3 with genus g and singularity basket Bp¤ˆ¤$+bûÿûÿûÿûÿ­˜¬Ì«Œ¬x¨%+__sig__6ª(ª@ª`ª€ªœ*RDestructively compute centres of X that have a plausible target in the sequence DB¤
¼
*bûÿûÿûÿûÿûÿŒª|ªlªpªÀŸ*lªxªûÿÿÿð¡*ÿÿÿÿÿÿÿÿtª¢*ZDestructively compute centres of K3s in the sequence DB that have a plausible target in DBô£*ûÿûÿûÿûÿûÿ®”ªlªˆª<¤*
__sig__1<\¥*lªªûÿÿÿ¼ô*codimies¦* `Compute a sequence containing the numbers of K3 surfaces in DB which might be unprojections of X Ð ü §*bûÿûÿûÿûÿûÿ´ª¤ªlª˜ª¸$¨*
__sig__2&ˆ&Ð&©*lª ªûÿÿÿ¤(ª*
Unprojections´+«*ÿÿÿÿÿÿÿÿœª¬*=Destructively compute unprojections of K3s in the sequence DBÌ4­*ûÿûÿûÿûÿûÿP®¼ªlª°ª:®*
__sig__3:;,;¯*lª¸ªûÿÿÿŒ<°*<A coded sequence of chains of unprojections within DB from X?t?±*bûÿûÿûÿûÿÌªlªÀªÄD²*
__sig__4EÌEäE³*lªÈªûÿÿÿ$H´*UnprojectionChainsüIJµ*ÿÿÿÿÿÿÿÿÄª¶*0A sequence of coded chains of projections from XP¨PàP·*bûÿûÿûÿûÿäªlªØª<T¸*
__sig__5UÜUV¹*lªàªûÿÿÿXÂ**‹True iff there is a centre of Type I among the known centres of X and all such centres have an image in codimension one less than that of X0`H`Ä*
__sig__7a@aXaÅ*lª«ûÿÿÿxbÆ*HasGoodTypeICentresÈdàdÇ*ÿÿÿÿÿÿÿÿ«È*-/usr/local/Magma2.8/package/AlgGeom/k3/hilb.m	ðŒ	É*-/usr/local/Magma2.8/package/AlgGeom/k3/hilb.m	ðŒ	Ê*>Degree of the polarising divisor on a virtual K3 with data g,BÄrË*ûÿûÿûÿûÿ`®4«$«(«DvÌ*
__sig__0wŒw¤wÍ*$«0«ûÿÿÿHyÎ*FýGiven integer g and basket of singularities B in the form [r,a] corresponding to the polarised quotient singularity 1/r(a,-a) return the Hilbert series of the corresponding K3 surface (whether it exists or not) polarised by a divisor with g + 1 sections ˆÏ*âÿûÿûÿûÿûÿl«D«$«8«ÐÐ*
__sig__1ÜôÑ*$«@«ûÿÿÿ˜’Ü*
__sig__0ipliedUpÒ* `Return a good candidate for the multiplied up Hilbert series, where g,B are as in Hilbert SeriesšØšðšÓ*âÿûÿûÿT«$«H«¸žÔ*
__sig__2 € ˜ Õ*$«P«ûÿÿÿx¢Ö*HilbertSeriesMultipliedUpÐ¤è¤×*ÿÿÿÿÿÿÿÿL«Ý*d«p«ûÿÿÿ€©J*
BaseComponentü:K*ÿÿÿÿÿÿÿÿ©L*KA complement in L to the space of sections which contain the hypersurface Sì?@M*ßûÿûÿûÿûÿ„å<©´¨0©„EN*
__sig__6FlF”FO*´¨8©ûÿÿÿtI2%
__sig__0.8/package/ô/DualFans,
__sig__1eIntersecti*SetAFRPP*=/usr/local/Magma2.8/package/AlgGeom/linearsystem/sub_linsys.m	à–	Q*=/usr/local/Magma2.8/package/AlgGeom/linearsystem/sub_linsys.m	à–	R*(‚The pullback of L by f. The second return value is a comparison map from the coefficient space of L to that of the pullback system^@^S*ûÿûÿûÿT©D©H©bV*"hThe subsystem of L for which the pullback by f contains the i-th coordinate hyperplane to multiplicity mfˆf fW*ûÿûÿûÿûÿd©D©X©HlX*
__sig__1m`mxmY*D©`©ûÿÿÿ¼oZ*=/usr/local/Magma2.8/package/AlgGeom/linearsystem/elt_linsys.m	à–	[*=/usr/local/Magma2.8/package/AlgGeom/linearsystem/elt_linsys.m	à–	\*4The linear space linearly spanned by the points of PzTz]*ÿíßûÿûÿûÿûÿŒ©|©l©p©4^*
__sig__0€¬€Ä€_*l©x©ûÿÿÿ†a*ÿÿÿÿÿÿÿÿt©b*;The linear space in the scheme S spanned by the subscheme T$Xc*ßßßûÿûÿûÿûÿ”©l©ˆ©È’d*
__sig__1“à“ø“e*l©©ûÿÿÿ@–f*NThe unique line in the projective space P through the points of S if it existsœg*ÿíêßûÿûÿûÿûÿÔ©¤©l©˜©X h*
__sig__2¡ ¡Ð¡i*l© ©ûÿÿÿP¤j*OThe unique conic in the projective plane P through the points of S if it exists¨(¨x¨>*
IsBasePointFreex“?*ÿÿÿÿÿÿÿÿì¨@*?True iff the hypersurfaces of L have no common geometric points ›¸›Ð›A*ûÿûÿûÿûÿ©´¨© B*
__sig__4¡@¡x¡C*´¨©ûÿÿÿ ¤E*ÿÿÿÿÿÿÿÿ©˜*Baskets@§À)RestrictionToPatch	pÀ	Á	Á)ÿÿÿÿÿÿÿÿô¦Â)"mThe restriction of f to a map from the ith affine patch of its domain to the jth affine patch of its codomain	øW	Ã)00ûÿûÿûÿûÿ§ì¦§l	Ä)
__sig__1n	Xn	ˆn	Å)ì¦§ûÿÿÿhm	Æ)2/usr/local/Magma2.8/package/AlgGeom/map/new_maps.m	€š	¸›	Ç)"2/usr/local/Magma2.8/package/AlgGeom/map/new_maps.m	Ðš	¸›	È)The identity map of X	¨	É)ßXûÿûÿûÿûÿ,§§ §ø	/usr/local/Magma2.8/libsJexamples:galpols:glnzgps:intro:isolgps:matgps:perfgps:pergps:simgps:solgps/tmp8$>Roly := fun:fÌç5hÌçÁä»ç

	












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__sig__0maliserge*˜¤ûÿÿÿatTç__sig__0t/Smash/larger.mÞ
__sig__0ßlxûÿÿÿã7/usr/local/Magma2.8/package/GrpMat/Smash/extraspecial.mð|	Ú1/usr/local/Magma2.8/package/GrpMat/Smash/larger.mð|	h†	á)/usr/local/Magma2.8/package/Code/isproj.m	@}	}	 é	ä»çÛ1/usr/local/Magma2.8/package/GrpMat/Smash/larger.m}	h†	Ü8ÄCompute kernel for homomorphism of semilinear G into cyclic group and write kernel over the larger field; return both representations, the cyclic group C and list of images in C of generators of GàWriteOverLargerField£<

magma/packages.mÖìÌç×ìüËçØìøËçáÿÿÿÿÿÿÿÿt&¹Ýjjj“ûÿ|lpâ7/usr/local/Magma2.8/package/GrpMat/Smash/charpol-test.mh†	€7/usr/local/Magma2.8/package/GrpMat/Smash/is-primitive.m(	‰
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__sig__0ä7/usr/local/Magma2.8/package/GrpMat/Smash/extraspecial.mh†	åMTrue iff G is known to normalise an extrapecial p-group or symplectic 2-groupæjûÿûÿûÿûÿ ”èœûÿÿÿéIsExtraSpecialNormaliserêÿÿÿÿÿÿÿÿ˜ë4/usr/local/Magma2.8/package/GrpMat/Smash/functions.m‚	h†	ì<4/usr/local/Magma2.8/package/GrpMat/Smash/functions.mƒ	h†	íThe block system of Gîj‹ûÿûÿûÿûÿÐÀ°´ð°¼ûÿÿÿñ
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__sig__1ö°Ôûÿÿÿ
__sig__8ters÷FReturn prime and exponent of the extraspecial subgroup normalised by Møjûÿûÿûÿûÿøè°Üù
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__sig__3°üûÿÿÿ%Return the extraspecial subgroup of Mjjûÿûÿûÿûÿ°
__sig__4°ûÿÿÿ%Return the extraspecial subgroup of Mzjûÿûÿûÿûÿ °
__sig__5°ûÿÿÿ	The tensor factors of M
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__sig__6°,ûÿÿÿ
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__sig__7°DûÿÿÿNReturn the change of basis matrix which exhibits the tensor decomposition of MzûÿûÿûÿûÿhX°L°TûÿÿÿØ
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	__sig__11&°”ûÿÿÿ'The centralising matrix of M(zûÿûÿûÿûÿ¸¨°œ*°¤ûÿÿÿ?TensorInducedFactorsØ+CentralisingMatrix,ÿÿÿÿÿÿÿÿ -The centralising matrix of G.jûÿûÿûÿûÿÀ°´/
	__sig__130°¼ûÿÿÿG
	__sig__18rss1 The Frobenius automorphisms of M2zûÿûÿûÿûÿàÐ°Ä3
	__sig__144°Ìûÿÿÿ5FrobeniusAutomorphisms6ÿÿÿÿÿÿÿÿÈ7 The Frobenius automorphisms of G8jûÿûÿûÿûÿè°Ü9
	__sig__15:°äûÿÿÿ<zûÿûÿûÿûÿø°ì=
	__sig__16>°ôûÿÿÿOTensorInducedPermutations(@ÿÿÿÿÿÿÿÿðA!Return the tensor induced factorsBjûÿûÿûÿûÿ°C
	__sig__17D°ûÿÿÿEReturn the tensor induced basisFzûÿûÿûÿûÿH °H°ûÿÿÿITensorInducedBasisJÿÿÿÿÿÿÿÿS
	__sig__20tationsKEReturn the permutations of the factors induced by the generators of MLzûÿûÿûÿûÿX8°,M
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__sig__9ïŒ¸ûÿÿÿð0/usr/local/Magma2.8/package/GrpMat/Smash/smash.mHÆ	h	;/usr/local/Magma2.8/package/GrpMat/Tensor/is-projectivity.m˜Æ	É	5/usr/local/Magma2.8/package/GrpMat/Tensor/is-tensor.m	É	5/usr/local/Magma2.8/package/GrpMat/Tensor/is-tensor.m	É	True iff G is a tensor productPjûÿûÿûÿûÿTD484@ûÿÿÿd
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IsolInfoMÿÿÿÿÿÿÿÿ$V
__sig__6eFieldN@The number of Isol groups of degree n over GF(p) in the databaseOûÿûÿûÿûÿD¼8P
__sig__5Q¼@ûÿÿÿc¼ˆûÿÿÿØeÿÿÿÿÿÿÿÿ„f?A group from the Isol Group database satisfying the predicate fÔìUjûÿûÿûÿûÿ\¼PW¼XûÿÿÿX
	IsolGroupYÿÿÿÿÿÿÿÿTZUVectors of the underlying vector space for G, in sequence suitable for use by Semidir[jûÿûÿûÿûÿt¼h\
__sig__7]¼pûÿÿÿvIsolGroupOfDegreeSatisfyingü$wÿÿÿÿÿÿÿÿÌŒ	__sig__15FieldSatisfying{¼èûÿÿÿ0jIsolGroupSatisfyingü$kÿÿÿÿÿÿÿÿœz
	__sig__12atisfyingn
	__sig__10°Ðb
x`HGiven: An irreducible matrix group G, and A specially ordered sequence Q of the vectors in RSpace(G). Return: The image H of the natural permutation representation of the 	semidirect product of G with its natural submodule generated by 	the vector (0, ..., 0, 1)gjûÿûÿûÿûÿ¤¼˜h
__sig__9i¼ ûÿÿÿxKThe groups of degree n in the Isol Group database satifying the predicate f$Lyûÿûÿûÿûÿì¼à‚BKLCarCodel@The groups in the Isol Group database satisfying the predicate fmûÿûÿûÿûÿ¼¼°o¼¸ûÿÿÿpIsolGroupsSatisfyingqÿÿÿÿÿÿÿÿ´rKA group of degree n from the Isol Group database satisfying the predicate fsjûÿûÿûÿûÿÔ¼Èt
	__sig__11u¼ÐûÿÿÿŽ
IsolProcess\„ÿÿÿÿÿÿÿÿ,WProcess to search through all Isol Groups in the database with either degree equal to n0H`/Subcodend‚ IsolGroupOfDegreeFieldSatisfyingŒ¤ƒÿÿÿÿÿÿÿÿü…ûÿûÿûÿûÿ¼ÀØð|IsolGroupsOfDegreeSatisfying}ÿÿÿÿÿÿÿÿä~VA group of degree n over GF(p) from the Isol Group database satisfying the predicate fjûÿûÿûÿûÿ¼ø€
	__sig__13¼ûÿÿÿ‘ûÿûÿûÿûÿ\L¼@L’
	__sig__16Ø
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“¼Hûÿÿÿœ”IsolProcessOfDegree¬Ä„YThe groups of degree n over GF(p) from the Isol Group database satisfying the predicate f†
	__sig__14‡¼ûÿÿÿˆ!IsolGroupsOfDegreeFieldSatisfying‰ÿÿÿÿÿÿÿÿŠ9Process to search through all Isol Groups in the database‹ûÿûÿûÿûÿ4¼(¼0ûÿÿÿªIsolProcessIsEmpty¬Ô«ÿÿÿÿÿÿÿÿœ¬'Extract current group from Isol processÔh®
	__sig__21H`¯¼¸ûÿÿÿžIsolProcessOfFieldü$Ÿÿÿÿÿÿÿÿÿl¨
	__sig__20eField¡ûÿûÿûÿûÿŒ¼€P•ÿÿÿÿÿÿÿÿD–\Process to search through all Isol Groups in the database with degree in the range specified—ûÿûÿûÿûÿd¼X˜
	__sig__17™¼`ûÿÿÿšQProcess to search through all Isol Groups in the database with field as specified›ûÿûÿûÿûÿt¼hœ
	__sig__18¼pûÿÿÿ­jûÿûÿûÿûÿ¼¼°L°IsolProcessGroup
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	__sig__22$ \Process to search through all Isol Groups in the database with degree and field as specified¢
	__sig__19£¼ˆûÿÿÿ¤IsolProcessOfDegreeField¥ÿÿÿÿÿÿÿÿ„¦FReturns true if the Isol Group process tuple has passed its last group§ûÿûÿûÿûÿ¤¼˜©¼ ûÿÿÿÂ
IsolProcessNext„ÃÿÿÿÿÿÿÿÿüÄ./usr/local/Magma2.8/package/GrpAb/direct_sum.m	èC	Å./usr/local/Magma2.8/package/GrpAb/direct_sum.m	èC	È
__sig__0Øð¶IsolProcessLabelü$·ÿÿÿÿÿÿÿÿÌ¸,Info string of current group of Isol process˜º
	__sig__23x»¼èûÿÿÿÀ²(Label of current group from Isol process³ûÿûÿÔ¼Èµ¼ÐûÿÿÿÆ&zThe direct sum of the abelian groups in S followed by the sequences of canonical inclusions and projections, respectively.(Çÿ““ûÿûÿ¨$$ôAGCodes¹ûÿûÿûÿûÿì¼à¼
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	__sig__24Á¼ûÿÿÿ:ÎThe (weakly) algebraic-geometric code build by evaluating functions of the Riemann-Roch space of D on the support S of places of degree 1 together with this basis. The degree of D need not be bounded by #S.
__sig__1xAGCode°â
HomomorphismsÔãÿÿÿÿÿÿÿÿ|ä bA sequence of all homomorphims from the finite abelian PC group G to the finite abelian PC group H¨Àæ
__sig__5¨Àó1/usr/local/Magma2.8/package/Code/alg_geom_codes.mK	P	õ

IsWeaklyAG8÷
DivisorüøGeometricSupport@ùRiemannRochBasisB€Y8ö
Ž‘úGoppaDesignedDistanceû8ÂGiven an integer q, builds the Hermitian code with support all places of degree one of the curve x^(q+1) + y^(q+1) + z^(q+1) over GF(q^2), except the place over (1:1:0), and divisor divisor r*P.:üHûÿûÿûÿûÿøÌì‡]M9ý
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HomGeneratorsÙÿÿÿÿÿÿÿÿTÛEcentral extension of (abelian) U by G determined by cocyclic matrix Am the ÚûÿûÿûÿûÿpÈylÝ
__sig__3ûÿûÿûÿûÿÜÈytûÿÿÿ__si…CentralExtensionpûÿÿÿÞ\A sequence of all homomorphims from the finite abelian group G to the finite abelian group Hß““ûÿûÿûÿûÿ”„,xà
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__sig__3cCode(ÿþHûÿûÿûÿûÿ(Ì$	__sig__2dŒ´åûÿûÿûÿûÿœ,ç,˜ûÿÿÿèOBSOLETE: Use Homomorphismsé““ûÿûÿûÿûÿ¼¬, ê
__sig__6ë,¨ûÿÿÿìAllHomomorphismsíÿÿÿÿÿÿÿÿ¤îOBSOLETE: Use HomomorphismsïûÿûÿûÿûÿÄ,¸ð
__sig__7ñ,Àûÿÿÿò1/usr/local/Magma2.8/package/Code/alg_geom_codes.mØW	h	4'/usr/local/Magma2.8/package/Code/BKLC.mP]	6RReturns the ith entry of length n and dimension k, which are non-negative integersd7HûÿûÿûÿûÿäÐØ8
__sig__0dB€Y89ÐàûÿÿÿF
BKLCUpperBoundˆGÿÿÿÿÿÿÿÿIqûÿûÿûÿûÿ,Ð J
__sig__3ÜôKÐ(ûÿÿÿdLBLLCUpperBoundÈì=qûÿûÿûÿûÿüÐð?Ðøûÿÿÿ”@
BKLCLowerBounddAÿÿÿÿÿÿÿÿôB$sReturns the upper bound on the MinimumDistance of a Linear Code of Length n and Dimension k over the finite field FCqûÿûÿûÿûÿÐ]S9D
__sig__2ÈEÐûÿÿÿ›ÿÿÿÿÿÿÿÿ\"œ!True if and only if G is a genus.x

¼ûÿûÿûÿûÿ|"`!p"” IsGenusT¡ÿÿÿÿÿÿÿÿt"¬Primesd¢(True if and only if G is a spinor genus. £¼ûÿûÿûÿûÿ”"`!ˆ"ð¤
	__sig__12(@¥`!"ûÿÿÿ˜¦
IsSpinorGenusÀ§ÿÿÿÿÿÿÿÿŒ"¨&The primes of the local genus symbols.ª
	__sig__13!0!…`!"ûÿÿÿà	†HackobjPrintSymGenLocø	‡ÿÿÿÿÿÿÿÿ"‰¼ûÿûÿûÿûÿûÿ,"`! "L
‹`!("ûÿÿÿÜŒHackobjPrintSymGend!&xReturn true if and only if C is of algebraic-geometric construction of length n, built from a divisor D with deg(D) < n.HûÿûÿûÿûÿXÌL
__sig__4ÌTûÿÿÿ'GeometricSupport°Ø(ÿÿÿÿÿÿÿÿ)TGiven an algebraic-geometric code, returns the divisor from which C was constructed.d*HûÿûÿûÿûÿT•°Ì¤xHûÿûÿûÿûÿpÌd
IsStronglyAGÿÿÿÿÿÿÿÿhOGiven an algebraic-geometric code, returns the curve from which is was defined. HéûÿûÿûÿûÿˆÌ|!
__sig__6"Ì„ûÿÿÿ# _Given an algebraic-geometric code, returns the sequence of places from which forms its support.$Hûÿûÿûÿûÿ˜ÌŒ%
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	CodeEntry`ˆ;ÿÿÿÿÿÿÿÿÜ<(†Returns the best known lower bound on the Maximal MinimumDistance of a Linear Code of Length n and Dimension k over the finite field FT>
__sig__1Œ´vBDLCdÈ+
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BestCodeance-"jGiven an algebraic-geometric code built with a divisor D, returns the Goppa designed distance: n - deg(D)..HûÿûÿûÿûÿÀÌ´/
__sig__90Ì¼ûÿÿÿ1GoppaDesignedDistance2ÿÿÿÿÿÿÿÿ¸3'/usr/local/Magma2.8/package/Code/BKLC.mh	LReturn the Elias asymptotic upper bound for delta in [0, 1] over the field KŽVqVûÿûÿûÿûÿ@4B€Y8
__sig__1H<ûÿÿÿ£GilbertVarshamovAsymptoticBound(¤ÿÿÿÿÿÿÿÿ€¥"kReturn the Griesmer lower bound of the length of a linear code of dimension k and minimum distance d over K$¦qûÿûÿûÿûÿ ”d§
__sig__5H`•
__sig__2d–Tûÿÿÿ—GilbertVarshamovBound˜ÿÿÿÿÿÿÿÿP™(Return the Gilbert-Varshamov lower bound of the size of a largest linear code of length n and minimum distance d over the field KY8šqûÿûÿûÿûÿpd{i8:›
__sig__3œlûÿÿÿGilbertVarshamovLinearBoundWèi;¡
__sig__4TT1Returns a Code of Dimenson k and MinimumDistance >=d over the finite field F with the minimum known possible Length. If the required code has Length >= 256 then it is not available and the Boolean value will return false (along with an irrelevant ZeroCode), otherwise the boolean return value will be true ÈHF÷Returns the best known upper bound on the minimum possible Length for which a Linear Code of Dimension k over the finite field F has MinimumDistance >= d. If the length required is > 256, then it is not available, and the value 0 will be returned Mÿÿÿÿÿÿÿÿ$N>ÛReturns the lower bound on the Length for which a Linear Code of Dimension k over the finite field F has MinimumDistance >= d. If the length required is > 256, then it is not available, and the value 0 will be returned OqûÿûÿûÿûÿDÐ8P
__sig__4QÐ@ûÿÿÿj
BDLCUpperBound„kÿÿÿÿÿÿÿÿœt
	__sig__10rCodeì
l$uReturns the Code of Length n and MinimumDistance d over the finite field F with the largest known possible Dimension.ðn
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	__sig__11x_ÿÿÿÿÿÿÿÿl`.™Returns the best known lower bound on the maximum possible Dimension for which a Linear Code of Length n over the finite field F has MinimumDistance >= d aqûÿûÿûÿûÿŒÐ€`XBestLengthLinearCodeYÿÿÿÿÿÿÿÿTZT1Returns a Code of Dimenson k and MinimumDistance >=d over the finite field F with the minimum known possible Length. If the required code has Length >= 256 then it is not available and the Boolean value will return false (along with an irrelevant ZeroCode), otherwise the boolean return value will be true[qHûÿûÿûÿtÐh]ÐpûÿÿÿmqHûÿûÿûÿûÿ¼Ð°LoÐ¸ûÿÿÿ(pBestDimensionLinearCodeÔqÿÿÿÿÿÿÿÿ´sqHûÿûÿûÿûÿÔÐÈ”b
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BDLCLowerBoundeÿÿÿÿÿÿÿÿ„f*ˆReturns the best known upper bound on the Dimension for which a Linear Code of Length n over the finite field F has MinimumDistance >= dgqûÿûÿûÿûÿ¤Ð˜h
__sig__8iÐ ûÿÿÿƒÿÿÿÿÿÿÿÿü„*/usr/local/Magma2.8/package/Code/BKLCdb2.mx~	¸	…(/usr/local/Magma2.8/package/Code/bound.mmÀ~	¸	†.(/usr/local/Magma2.8/package/Code/bound.m	¸	ˆqûÿûÿûÿûÿ(ðwÿÿÿÿÿÿÿÿÌx$sReturns the Best Known Linear Code (of Maximal MinimumDistance) of Length n and Dimension k over the finite field F0yqHûÿûÿûÿûÿìÐàD{Ðèûÿÿÿ¨Àr$uReturns the Code of Length n and MinimumDistance d over the finite field F with the largest known possible Dimension.uÐÐûÿÿÿ‡$vReturn the upper Elias bound for the cardinality of a largest code of length n and minimum distance d over the field Kð
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EliasBound|”Œÿÿÿÿÿÿÿÿ ¨œûÿÿÿô|BestKnownLinearCode}ÿÿÿÿÿÿÿÿä~$sReturns the Best Known Linear Code (of Maximal MinimumDistance) of Length n and Dimension k over the finite field FqHûÿûÿûÿûÿÐø€
	__sig__12ÐûÿÿÿžÿÿÿÿÿÿÿÿhŸ$pReturn the Gilbert-Varshamov asymptotic lower bound of the information rate for delta in [0, 1] over the field Kè VqVûÿûÿûÿûÿˆ|¢„ûÿÿÿ`³
__sig__7ightBound‘EliasAsymptoticBound$’ÿÿÿÿÿÿÿÿ8“(‚Return the Gilbert-Varshamov lower bound for the cardinality of a largest code of length n and minimum distance d over the field K¨À”qûÿûÿûÿûÿXLð,bhGiven sequences alpha = [a_1, ..., a_n], W = [w_1, ..., w_s], and Z = [z_1, ... z_k] of elements from the extension field K of the finite field S, such that the elements of alpha and Z are non-zero and the n + s elements of alpha and W are distinct, together with an integer t > 0, construct the generalized Srivastava code of parameters alpha, W, Z, t, over SˆGeneralizedSrivastavaCodeÄÿÿÿÿÿÿÿÿHV:Given sequences alpha = [a_1, ..., a_n] and W = [w_1, ..., w_s] of elements from the extension field K of the finite field S, such that the elements of alpha are non-zero and the n + s elements of alpha and W are distinct, together with an integer mu, construct a Srivastava code of parameters alpha, W, mu, over SC€Y8qÿrÿrHûÿûÿûÿûÿh@\d@dûÿÿÿ
SrivastavaCodeC€Y8ÿÿÿÿÿÿÿÿ`@áGiven alpha = [a_1,...a_n], and w = [w_1,...w_s] a sequence of n+s distincts elements of GF(q), z, a sequence of n nonzero elements of GF(q), and t > 0, construct the Gabidulin MDS code of parameters alpha, w, z, t over GF(q)“PadCodeodeµ
GriesmerBound„¶ÿÿÿÿÿÿÿÿÈ·"nReturn the Johnson upper bound for the cardinality of a largest binary code of length n and minimum distance dÀ¸ûÿûÿûÿûÿèÜ¹
__sig__8ØðÉEntropy©GriesmerLengthBoundü$ªÿÿÿÿÿÿÿÿ˜«$tCompute the Griesmer upper bound of the minimum weight of a linear code of length n and dimension k over the field KP¬qûÿûÿûÿûÿ¸¬­
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JohnsonBoundœ¼ÿÿÿÿÿÿÿÿà½CReturn true if the binary code is nearly perfect, else return falseÜô¾Hûÿûÿûÿûÿô4¿
__sig__90®´ûÿÿÿ¯GriesmerMinimumWeightBound°ÿÿÿÿÿÿÿÿ°±"mReturn the Griesmer upper bound of the cardinality of a linear code of length n and minimum distance d over K²qûÿûÿûÿûÿÐÄ´ÌûÿÿÿÍMcElieceEtAlAsymptoticBoundü$Îÿÿÿÿÿÿÿÿ(Ï$pReturn the Plotkin upper bound of the cardinality of a largest code of length n and dimension d over the field K8PÐqûÿûÿûÿûÿH<ÒDûÿÿÿ¼Á
IsNearlyPerfectÔÂÿÿÿÿÿÿÿÿøÃ&wReturn the Levenstein upper bound of the cardinality of a largest code of length n, minimum distance d over the field K 8PÄqûÿûÿûÿûÿÅ
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PlotkinBoundˆÔÿÿÿÿÿÿÿÿ@Õ9Return true if the code is equidistant, else return false„œÖHûÿûÿûÿûÿ`Td×
	__sig__13 ÈØ\ûÿÿÿôÙIsEquidistantð ÆûÿÿÿÇLevenshteinBoundÈÿÿÿÿÿÿÿÿÑ
	__sig__12oticBoundÉ$tReturn the McEliece-Rodemich-Rumsey-Welch asymptotic upper bound of the binary information rate for delta in [0, 1].ÊVVûÿûÿûÿûÿ0$Ë
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ç"mReturn the Singleton asymptotic upper bound of the information rate for delta in [0, 1] over any finite fieldÀèVVûÿûÿûÿûÿ¨œðé
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	__sig__15ound¬Û fReturn the Plotkin asymptotic upper bound of the information rate for delta in [0, 1] over the field KÀÜVqVûÿûÿûÿûÿxlÝ
	__sig__14ØðÞtûÿÿÿ4Lê¤ûÿÿÿØëSingletonAsymptoticBound$Lìÿÿÿÿÿÿÿÿ íJComputes the volume of the closed ball of radius r in the vector space K^n|”îqûÿûÿûÿûÿÀ´Äð¼ûÿÿÿðßPlotkinAsymptoticBoundàÿÿÿÿÿÿÿÿpá$vReturn the Singleton upper bound of the cardinality of a largest code of length n, minimum distance d over the field Kâqûÿûÿûÿûÿ„äŒûÿÿÿýSpherePackingBound¬Ôþÿÿÿÿÿÿÿÿèÿ5Return true if C is a perfect code, else return false˜Hûÿûÿûÿûÿü`
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SphereVolumeÔòÿÿÿÿÿÿÿÿ¸ó*‡Return the Hamming sphere packing upper bound of the cardinality of a largest codes of length n and minimum distance d over the field K¨ÀôqûÿûÿûÿûÿØÌðõ
	__sig__18Ôì>¨ôûÿÿÿØ?SubcodeWordsOfWeight@ÿÿÿÿÿÿÿÿðAHGiven a set of non-negative integers S, each less than Length(C), return the subcode generated by those words of C whose weights lie in S If Cutoff is set to a non-zero positive integer then the number of words used of each weight will be limited to Cutoff.h€C__sig__4pˆ ÀöÔûÿÿÿ÷SpheresPackingBoundøÿÿÿÿÿÿÿÿÐù*‡Return the Hamming sphere packing upper bound of the cardinality of a largest codes of length n and minimum distance d over the field Kúqûÿûÿûÿûÿðäû
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VanLintBound„ÿÿÿÿÿÿÿÿ('/usr/local/Magma2.8/package/Code/cons.m˜«	'/usr/local/Magma2.8/package/Code/cons.m˜«	
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__sig__0¤¼@LûÿÿÿÜô fReturn the Hamming asymptotic upper bound of the information rate for delta in [0, 1] over the field KVqVûÿûÿûÿûÿûÿÿÿHammingAsymptoticBoundÿÿÿÿÿÿÿÿ	&zReturn the "Van Lint" lower bound to the size of a largest linear code of length n and minimum distance d over the field K
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__sig__0dB€Y8.¨´ûÿÿÿ0ÿÿÿÿÿÿÿÿ°1,Returns the subcode of C generated by the basis elements numbered in S, which must contain positive integers less than or equal to Dimension(C)B€Y82ÿHHûÿûÿûÿûÿÐ¨Ä3c-:3
__sig__1È4¨Ìûÿÿÿ5 bGiven a linear code C1 and a subcode C2 of C1, return a subcode of C1 of dimension k containing C26HHHûÿûÿûÿûÿà¨Ô„7
__sig__2d8¨ÜûÿÿÿPÿÿÿÿÿÿÿÿ0QITake the residue of code C by puncturing at the support of a minimum word¨RHHûÿûÿûÿûÿP¨D”S
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ResidueCodeH`VÿÿÿÿÿÿÿÿHY
__sig__8è9SubcodeBetweenCode;D¨ûÿÿÿ(ÿÿÿÿÿÿÿÿ)0/usr/local/Magma2.8/package/Code/constructions.m
¶	¸·	*40/usr/local/Magma2.8/package/Code/constructions.m`¶	¸·	+$qReturns a subcode of C dimension k, which must be a non negative integer less than or equal to the dimension of Cx,HHûÿûÿûÿûÿÈ¸¨¬#ZMGiven polynomials P and G over a finite field F, an integer n > 1, together with a subfield S of F, such that n is coprime to the cardinality of S, F is the splitting field of x^n - 1 over S, P and G are coprime to x^n - 1 and both have degree less than n, construct the Chien-Choy generalised BCH code with parameters P, G, n over S&@”ûÿÿÿ:ÿÿÿÿÿÿÿÿØ;>ÜGiven a non negative integer S less than Length(C), return the subcode generated by those words of C with weight W. If Cutoff is set to a non-zero positive integer then the number of words used will be limited to Cutoff.<HHûÿûÿûÿûÿø¨ì¼=
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__sig__5ght¼Ä*/usr/local/Magma2.8/package/Code/entropy.m@»	Á	ÅUComputes the entropy function for a real number in the interval [0,1-1/q], where q=#SY8ÆqVVûÿûÿûÿûÿ dÇ
__sig__0€Y8ÈûÿÿÿÄý&:â)/usr/local/Magma2.8/package/Code/isproj.m¸¼	ø¾	ã(Return true iff the code C is projective
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äHûÿûÿûÿûÿìP˜ˆŒôå
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IsProjectiveÄèÿÿÿÿÿÿÿÿé'/usr/local/Magma2.8/package/Code/kraw.mø¾	ê'/usr/local/Magma2.8/package/Code/kraw.mø¾	 é	ä»çÊÿÿÿÿÿÿÿÿË+Return the information rate of a given codeÜÌHûÿûÿûÿûÿ8,Í
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InformationRateÐÿÿÿÿÿÿÿÿ0Ñ'/usr/local/Magma2.8/package/Code/fire.mÁ	KHolesemI
CodeComplement„JÿÿÿÿÿÿÿÿK&xGiven codes C1=[n1,k,d1] and C2=[n2,k,d2], construct a [n1+n2,k,d>=d1+d2] code by pasting the generator matrix together  LHHHûÿûÿûÿûÿ8¨,`M
__sig__6D\N¨4ûÿÿÿ|O
Juxtaposition„BÿHHûÿûÿûÿûÿ¨E=Given a subcode C1 of C, returns a code C2 such that C=C1+C2.FHHHûÿûÿûÿûÿ ¨H¨ûÿÿÿa
ConstructionX3„bÿÿÿÿÿÿÿÿxc>×C2 and C3 must be subcodes of C1, while the dimensions of D2 and D3 must add with the dimensions to C2 and C3 to give the dimension of C1. The returned code is constructed via the constructionXX algorithm of Alltop.0dHHHHHHûÿûÿûÿûÿ˜¨Œ¬e
	__sig__10œ¼W"iGiven a code C1=[n,k,d], a subcode C2=[n,k-l,d+e], and a code C3=[a,l,e], construct a code C=[n+a,k,d+e].Ô
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XHHHHûÿûÿûÿûÿh¨\ðZ¨dûÿÿÿ`[
ConstructionXÄ\ÿÿÿÿÿÿÿÿ`^HHHHHHûÿûÿûÿûÿ€¨tü$<f¨”ûÿÿÿØg
ConstructionXXœhÿÿÿÿÿÿÿÿi,The direct sum (diagonal joining) of X and Ydjïÿïÿïÿûÿûÿûÿûÿ¸°¨¤xk
	__sig__11|”l¨¬ûÿÿÿÀmCThe direct sum (diagonal joining) of the matrices in the sequence Q(@Xp]<ÐGiven a chain of codes C1=[n,k1,d1], C2=[n,k2,d2], and C3=[n,k3,d3] with k3 < k2 < k1, and suffix codes D1=[n1,k1-k2,e1] and D2=[n2,k3-k2,e2], construct a code C=[n+n1+n2,k1,d] with d >= min{d3,d1+e1,d2+e2.} _
__sig__9`¨|ûÿÿÿ‰ExpurgateWeightCode¬ÔŠÿÿÿÿÿÿÿÿ‹Extend C n times.<dŒHHûÿûÿûÿûÿ8¨,P
	__sig__18ˆ°Ž¨4ûÿÿÿ0JGiven a linear code, lengthen it by n positions (adding columns of zeros).”¬‘
	__sig__19œ¼—
	__sig__20Ôìu
ZinovievCodeÔvÿÿÿÿÿÿÿÿÈw2«Given a binary linear [n,k,d] code C with dual distance d',return the [n-d',k-d'+1,>=d] code obtained by shortening in the positions of a word in Dual(C) of minimal weightüxHHûÿûÿûÿûÿøè¨Ü¼y
	__sig__14”¬z¨äûÿÿÿünÿïÿïÿûÿûÿûÿûÿÈÀ¨´o
	__sig__12p¨¼ûÿÿÿq[Construct a generalized concatenated code from the given sequences of inner and outer codesrÿHÿHHûÿûÿûÿûÿÐ¨Äs
	__sig__13t¨ÌûÿÿÿHHûÿûÿûÿûÿH¨<L’¨Dûÿÿÿ(”ÿÿÿÿÿÿÿÿ@•8ÁIf C is a code over F, then each element of F is mapped into a column vector over K. Returned is the code generated when each codeword of C is mapped into multiple codewords over K in this way.8P˜¨\ûÿÿÿˆ™SubfieldCode Øð{
ConstructionY1|ÿÿÿÿÿÿÿÿà}$uShorten the linear code C at the support of a word of weight w of the dual code, resulting in an [n-w,k-w+1,>=d] code~HHûÿûÿûÿûÿ¨ô
	__sig__15€¨üûÿÿÿOThe code obtained by deleting the code words contained in the sequence L from C‚ÿuHHûÿûÿûÿûÿ¨ƒ
	__sig__16„¨ûÿÿÿ…1Delete a subspace generated by a word of weight w†HHûÿûÿûÿûÿ ¨‡
	__sig__17ˆ¨ûÿÿÿ©SubfieldRepresentationCodeü$ªÿÿÿÿÿÿÿÿ˜«0¡Create the [Nn,Kk,delta >= dD] concatenated code associated with the Outer [N,K,D]-code over GF(q^k) and the Inner [n,k,d]-code over GF(q) (where q is arbitrary)ü¬HHHûÿûÿûÿûÿ¸¨¬¼®¨´ûÿÿÿÜ¯ConcatenatedCode$<³
	__sig__25d|šÿÿÿÿÿÿÿÿX¤¨Œûÿÿÿ›<ÏGiven a linear code over GF(q^m), return a code whose codewords are obtained from those of C by expanding each coordinate in GF(q^m) as a vector of dimension m+1 over K := GF(q), including a parity check bit0H`œqHHûÿûÿûÿûÿx¨lÜ
	__sig__21äü£
	__sig__22$L–qHHûÿûÿûÿûÿ`¨T°ÿÿÿÿÿÿÿÿ°±-The direct sum of the codes in the sequence QÀ
²ÿHHûÿûÿûÿûÿDÐ¨ÄÄ´¨Ìûÿÿÿôµ+/usr/local/Magma2.8/package/Code/decodeml.m˜Ý	 Þ	¶+/usr/local/Magma2.8/package/Code/decodeml.màÝ	 Þ	·>Return a maximal set of equations orthogonal for each positionph
rceSymGenž¨tûÿÿÿŸ SubfieldRepresentationParityCode ÿÿÿÿÿÿÿÿp¡GReturn the code concatenation of codes C1 and C2 (over the same field) ¢HHHûÿûÿûÿûÿ¨„­
	__sig__24tionCode¥4°Given a linear code over GF(q^m), return a code whose codewords are obtained from those of C by expanding each coordinate in GF(q^m) as a vector of dimension m over K := GF(q) ¦qHHûÿûÿûÿûÿ ¨”§
	__sig__23¨¨œûÿÿÿÒ'/usr/local/Magma2.8/package/Code/fire.mä	Ó*ŽGiven a polynomial h in GF(q)[X], a nonnegative integer s, constructs a Fire code of length n with generator polynomial h*(X^s - 1) over GF(q)ÔHûÿûÿûÿûÿXHLÄÕ
__sig__0¨ÀÖHTûÿÿÿì×
FireCodeÄÜØÿÿÿÿÿÿÿÿPÝ
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MaxOrthPCheckÔ¼ÿÿÿÿÿÿÿÿà½<ÓDecoder for binary codes. Given a list of received vectors, return a two sequences. The second contains the corrected vectors, the first, boolean values saying if the decoder could retrieve the original codeword0¾ÿuHûÿûÿûÿØô¬¿
__sig__1œ¼ÀØüûÿÿÿL¸HûÿûÿûÿûÿèØÜ¹
__sig__0ºØäûÿÿÿÙ(/usr/local/Magma2.8/package/Code/ispow.møæ	0é	Ú(/usr/local/Magma2.8/package/Code/ispow.m
@ç	0é	ÛDReturns whether y = x^n for some integer n, and if so also returns nÄÜûÿûÿûÿxhlÞhtûÿÿÿ0ß
	IsPowerOf àÿÿÿÿÿÿÿÿph$(/usr/local/Magma2.8/package/Plane/geom.mals.m	ä»çÁ
DecodeMLÂÿÿÿÿÿÿÿÿøÃ*/usr/local/Magma2.8/package/Code/entropy.m˜é	h	÷$oReturn the Krawchouk transform of the polynomial a over the rational field with respect to the vector space K^n¼
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øqûÿûÿûÿûÿè¨Üðú¨äûÿÿÿ`ûKrawchoukTransformèüÿÿÿÿÿÿÿÿàý&wReturn the inverse Krawchouk transform of the polynomial A over the rational field with respect to the vector space K^n|¤¼_˜™šïBinomialPolynomialü$ðÿÿÿÿÿÿÿÿ°ñQReturn the Krawchouk polynomial of parameter k and n in K over the Rational FieldxòqûÿûÿûÿûÿÐ¨ÄÄó
__sig__1¨Àô¨Ìûÿÿÿì3Return the Lee distance between the vectors v and wt99ûÿûÿûÿûÿ8,l
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LeeDistanceÿÿÿÿÿÿÿÿ0*/usr/local/Magma2.8/package/Code/mattsol.mHï	(ð	*/usr/local/Magma2.8/package/Code/mattsol.mï	(ð	
__sig__1nsformÀ
__sig__0ØðõKrawchoukPolynomialpˆöÿÿÿÿÿÿÿÿÈù
__sig__2 ÀëRGiven a polynomial P over the Rational Field, return the polynomial Binomial(P, l)0ìûÿûÿûÿûÿ¸¨¬¬í
__sig__0œ¼î¨´ûÿÿÿL(qHûÿûÿûÿûÿ¨€œL*€¤ûÿÿÿ(+)/usr/local/Magma2.8/package/Code/simple.mˆò	Xô	,
)/usr/local/Magma2.8/package/Code/simple.mÐò	Xô	-#Return the [n,0,n] zero code over RLd.ýÿHûÿûÿûÿûÿÀ°´à/
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ZeroCode À,Given A, a polynomial over a finite field containing a primitive n-th root of unity, return the inverse Mattson-Solomon transform of parameter nHlûÿÿÿŒ2ÿÿÿÿÿÿÿÿ¸3'Return the [n,n,1] universe code over R
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4ýÿHûÿûÿûÿûÿØ°ÌÄ5
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UniverseCode”8ÿÿÿÿÿÿÿÿÐ9#Return the [n,1,n] zero code over R À:ýÿHûÿûÿûÿûÿð°äðInverseMattsonSolomonTransformÿÿÿÿÿÿÿÿh)/usr/local/Magma2.8/package/Code/random.m ø	xø	InverseKrawchouk¬Ôÿÿÿÿÿÿÿÿø&/usr/local/Magma2.8/package/Code/lee.m	Ðú	&/usr/local/Magma2.8/package/Code/lee.m	Ðú	%Return the Lee weight of the vector v°9ûÿûÿûÿûÿ Œ
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	LeeWeightÔì
ÿÿÿÿÿÿÿÿþqýÿûÿûÿûÿûÿ¨ôÿ
__sig__3¨üûÿÿÿ )/usr/local/Magma2.8/package/Code/random.mû	øý	!HReturn a random linear code of length n and dimension k over the field K¼Ô"qHûÿûÿûÿûÿ €„À$€Œûÿÿÿt%RandomLinearCode¨À&ÿÿÿÿÿÿÿÿˆ'HReturn a random linear code of length n and dimension k over the field K$L)
__sig__1\|^Prime–*ˆGiven a, a polynomial over a finite field containing a primitive n-th root of unity, return the Mattson-Solomon transform of parameter n¼ÔûÿûÿûÿûÿXHLÀHTûÿÿÿtMattsonSolomonTransformÀÿÿÿÿÿÿÿÿP#__sig__0omonTransform„ûÿûÿûÿûÿpHd¼L*/usr/local/Magma2.8/package/Code/simplex.m@	À	M0Return the [2^r-1,r,2^(r-1)] binary simplex code
$NHûÿûÿûÿûÿ@04 O
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SimplexCodeØðRÿÿÿÿÿÿÿÿ8S0/usr/local/Magma2.8/package/Code/specificcodes.m	À	T0/usr/local/Magma2.8/package/Code/specificcodes.mP	À	‡MDSCodeeB°ûÿÿÿ$C
ZeroSumCodeðDÿÿÿÿÿÿÿÿE0Return the [n,n-1,2] even-weight code over GF(2)¨ÀFHûÿûÿûÿûÿ °G
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EvenWeightCodeJÿÿÿÿÿÿÿÿK*/usr/local/Magma2.8/package/Code/simplex.mÐ	0	;
__sig__2Ì<°ìûÿÿÿUVReturns the p-th power residue code of length n, |K| must be a p-th power residue of n°VqHûÿûÿûÿûÿ`PT4W
__sig__0$XP\ûÿÿÿ´YPowerResidueCode4ZÿÿÿÿÿÿÿÿX\HûÿûÿûÿûÿˆxPlp]
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RepetitionCodeÄ>ÿÿÿÿÿÿÿÿè?)Return the [n,n-1,2] zero-sum code over Rô@ýÿHûÿûÿûÿûÿ°ü<A__sig__3<\t”wZConstruct a binary "twisted QR" code of length l*m. Requires that l*m must be coprime to 2ÜüxHûÿûÿûÿûÿèPÜ`{
TwistedQRCodeP|ÿÿÿÿÿÿÿÿà…
__sig__8nant}"lConstructs a non-primitive alternant code of length n over GF(2), with n - mr <= k <= n - r, and d >= r + 1.ü~HûÿûÿûÿûÿPô$
__sig__7$D€Püûÿÿÿrculq:ËConstruction of a bordered doubly circulant linear binary codes C=[2p+1,p+1] based on quadratic residues modulo p. The first p rows are extended by a, the last row by b. p must be a prime greater than 2.ØðrþÿþÿHûÿûÿûÿûÿÐPÄ,tPÌûÿÿÿLuBorderedDoublyCirculantQRCode¼vÿÿÿÿÿÿÿÿÈy
__sig__6ÔìzPäûÿÿÿDd|[:ÎConstructs the quasi-cyclic code with generator polynomials given by Gen of length n. Attaches the Generator matrices of the cyclic codes generated by Gen side by side. Requires that n be a multiple of #Gen_
QuasiCyclicCode`ÿÿÿÿÿÿÿÿpaJConstructs the quasi-cyclic code with generator polynomials given by Gen of length n. Attaches the Generator matrices of the cyclic codes generated by Gen 2-Dimensionally, stacking with height h. Requires that #Gen be a multiple of h, and that n be a multiple of #Gen/hlbHûÿûÿûÿûÿP„ôc
__sig__2`xdPŒûÿÿÿÄe=Given an integer n, return the CordaroWagner code of length nàfHûÿûÿûÿûÿ P”g
__sig__3(hPœûÿÿÿ@	iCordaroWagnerCodeü$jÿÿÿÿÿÿÿÿ˜s
__sig__5Codedk(„Construction of doubly circulant linear binary codes C=[2p,p] based on quadratic residues modulo p. p must be a prime greater than 2DlHûÿûÿûÿûÿ¸P¬ìm
__sig__4ÄÜnP´ûÿÿÿ<oDoublyCirculantQRCode¼pÿÿÿÿÿÿÿÿ°•SymmetricWeightEnumerator–ÿÿÿÿÿÿÿÿ@—%/usr/local/Magma2.8/package/Code/z4.m	H	˜%/usr/local/Magma2.8/package/Code/z4.m	H	™8Return the factors of (x^n - 1) over Z_4, where n is oddšûÿûÿûÿûÿp`dÀœ`lûÿÿÿ°HûÿûÿûÿûÿÈ`¼H±
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²`Äûÿÿÿ˜³DThe standard form S of C together with the isomorphism map f: C -> Sð´HHûÿûÿûÿØ`Ì,µ
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StandardForml¸ÿÿÿÿÿÿÿÿÐ¹"Return the Gray map for the code C$D»__sig__64TtŒZ4CyclotomicFactorsžÿÿÿÿÿÿÿÿhŸFThe quaternary Kerdock Code K(m) defined by the primitive polynomial h< Hûÿûÿûÿûÿ˜ˆ`|¢`„ûÿÿÿd£
KerdockCode¤ÿÿÿÿÿÿÿÿ€¥KThe quaternary Kerdock Code K(m), defined by a default primitive polynomial§
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JustesenCode„Žÿÿÿÿÿÿÿÿ(*/usr/local/Magma2.8/package/Code/weights.m°	ð	”`@ûÿÿÿage/›
__sig__0umeratorä
__sig__1rmsorver Ô@The Hamming weight enumerator of the Z_4 code C (in 2 variables)s a polynomÕHKûÿûÿûÿûÿ\`PûÿÖ
	__sig__12	,¼ç×`Xûÿÿÿ@¡
__sig__1NonPrimitiveAlternant$‚ÿÿÿÿÿÿÿÿøƒ eGiven a finite field GF(q = 2^m), constructs the [q + 1,k,q - k + 2] Maximum Distance Seperable code.ð„qHûÿûÿûÿûÿP0†Pûÿÿÿ\ˆÿÿÿÿÿÿÿÿ‰ _Build a binary linear Justesen code of length 2mN and dimension mK where N=2^m-1 and 1 < K < N |œ¼‹
__sig__9ÄÜã‰ûÿûÿûÿûÿh3”lˆLålûÿÿÿ(æAThe class number of the binary quadratic forms of discriminant D.Üôçûÿûÿûÿûÿ¬¤l˜4è
__sig__20él ûÿÿÿPê5The number of the binary quadratic form classes in Q.pë¨ûÿûÿûÿûÿàD´l¨˜ì
__sig__3¨Àíl°ûÿÿÿ8LŠHûÿûÿûÿûÿ0P$ŒP,ûÿÿÿ¦Hûÿûÿûÿûÿ `”H¨`œûÿÿÿ$©HThe quaternary Preparata Code P(m) defined by the primitive polynomial h ªHûÿûÿûÿûÿÀ°`¤`«
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PreparataCode„®ÿÿÿÿÿÿÿÿ¨¯MThe quaternary Preparata Code P(m), defined by a default primitive polynomial|îThe ideal class number of R.hïûÿûÿûÿûÿÄl¸ð
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PicardNumberóÿÿÿÿÿÿÿÿ¼ô3/usr/local/Magma2.8/package/QuadBin/reduced_forms.m¸(	¨.	õ3/usr/local/Magma2.8/package/QuadBin/reduced_forms.m)	¨.	ö\A sequence of representative reduced forms for the quadratic form classes of discriminant D.S':÷ûÿûÿûÿûÿôäÔØø
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ReducedFormsûÿÿÿÿÿÿÿÿÜü1A sequence of representative reduced forms for Q.,	ý¨ûÿûÿûÿûÿüÔðþ
__sig__1àÿÔøûÿÿÿRThe sequence of reduced orbits of binary quadratic forms of positive discriminant.;¨ûÿûÿûÿûÿÔì…_;
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ReducedOrbitsÿÿÿÿÿÿÿÿ4/usr/local/Magma2.8/package/QuadBin/abelian_groups.m-	¨.	2/usr/local/Magma2.8/package/QuadBin/class_groups.mmð-	¨.	2/usr/local/Magma2.8/package/QuadBin/class_groups.m@.	¨.	¾ÿÿÿÿÿÿÿÿè¿OReturn the image of the Z_4 code C under the Gray map, as a sequence of vectors¤¼ÔÀHûÿûÿûÿûÿ`üÀÁ
__sig__7,DÂ`ûÿÿÿÃ
GrayMapImageŒÄÿÿÿÿÿÿÿÿÅTGiven a Z_4 vector v, return the image of under the Gray map, as a vector over GF(2)|Æ9uûÿûÿûÿûÿ `TºHûÿûÿûÿûÿð`ä¼`ìûÿÿÿÍHasLinearGrayMapImageÔÎÿÿÿÿÿÿÿÿ(ÏBThe symmetric weight enumerator of the Z_4 code C (in 3 variables)èÐHKûÿûÿûÿûÿH`<Ñ
	__sig__100Ò`Dûÿÿÿ\’<The Lee weight enumerator of the Z_4 code C (in 2 variables)„‘HKûÿûÿûÿûÿP`H¼“LeeWeightEnumeratorôÓÿÿÿÿÿÿÿÿDÇ
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	__sig__11meratorÉ$uReturn true iff the image of the Gray map of the Z_4 code C is a linear GF(2) code; if so, return also the image codeÊHHûÿûÿûÿ0`$Ë
__sig__9Ì`,ûÿÿÿØHammingWeightEnumeratorÔÙÿÿÿÿÿÿÿÿTÚ2/usr/local/Magma2.8/package/QuadBin/class_number.m7	h9	Û
2/usr/local/Magma2.8/package/QuadBin/class_number.mX7	h9	Ü$The class of binary quadratic forms.ÄÝ¨ûÿûÿûÿûÿ|lpLÞ
__sig__04TßlxûÿÿÿœàBinaryQuadraticForms<áÿÿÿÿÿÿÿÿtâForm coefficients.4T	;The sequence of invariants of the abelian class group of Q.dœ
¨ûÿûÿûÿûÿ0 $œ
__sig__0|” ,ûÿÿÿÄ?The abelian class group of Q, followed by the isomorphism to Q.”¬Ä¨“ûÿûÿûÿÐD@ 4X
__sig__1Xp <ûÿÿÿWClass group of non maximal quadratic orders and a map from the group to the ideals of R 8X__sig__2d„Ìì/TwoTorsionSubgroup¬Ô0ÿÿÿÿÿÿÿÿ°14Returns true iff n is the order of the form class f.˜2‰ûÿûÿûÿûÿÐxÄ`3
__sig__3œÄ4xÌûÿÿÿð5$qA pseudo-random reduced form in Q; increasing the parameter Lambda achieves a better approximation to randomness.„œ6¨‰ûÿûÿûÿûÿäràxÔÜ7
__sig__4Ôì8xÜûÿÿÿLÐDensityŒ“ûÿûÿûÿP D LûÿÿÿÆþ;
PicardGroup@ÿÿÿÿÿÿÿÿH,The kernel of the fundamental quotient of Q.¨“ûÿûÿûÿh \3c-:
__sig__3È dûÿÿÿ95/usr/local/Magma2.8/package/QuadBin/binary_lattices.m	D	:5/usr/local/Magma2.8/package/QuadBin/binary_lattices.m	D	;+The Lattice of the binary quadratic form f.|”<‰ÔûÿûÿûÿûÿdøèìÈ=
__sig__0¬Ä>èôûÿÿÿðG
__sig__2berà?,The nth representation number of the form f.ð@‰ûÿûÿûÿûÿèüA
__sig__1 Bèûÿÿÿ„FundamentalKernelÿÿÿÿÿÿÿÿ`Æ+/usr/local/Magma2.8/package/FldFun/places.m é	ä»ç7/usr/local/Magma2.8/package/QuadBin/torsion_subgroups.m€I	
7/usr/local/Magma2.8/package/QuadBin/torsion_subgroups.m€I	&The Sylow p-subgroup of reduced forms. ¨“ûÿûÿûÿH€ˆx|!
__sig__0"x„ûÿÿÿ#
	pSubgroup$ÿÿÿÿÿÿÿÿ€%CThe sequence of reduced 2-torsion forms of negative discriminant D.(&¨ûÿûÿûÿûÿ x”$'
__sig__1d(xœûÿÿÿ)
AmbiguousFormsô*ÿÿÿÿÿÿÿÿ˜+(The subgroup of form classes of order 2.Xp,¨“ûÿûÿûÿX€¸x¬˜-
__sig__2¨À.x´ûÿÿÿ˜
OrthogonalSum„™ÿÿÿÿÿÿÿÿTš/The orthogonal direct sum of the lattices in Q.Ôh›ÿÔÔûÿûÿûÿûÿÜNtŒhðœ
	__sig__13\tŒpûÿÿÿÀž/The orthogonal direct sum of the lattices in Q.|”¬ŸÿÔÔûÿûÿûÿûÿ„Œx¼ 
	__sig__14Ôì¡Œ€ûÿÿÿD¢(The tensor product of lattices L1 and L24L¤
	__sig__15¤¼v"nReturns true if and only if the lattice has coordinates and inner product specified in the exact rings Z or Q.ì
x
__sig__5hyŒàûÿÿÿz)The ring of coefficients of the lattice Lˆ°{ÔýÿûÿûÿûÿûÿôŒèŒ|
__sig__6d|}Œðûÿÿÿ¼~#The quadratic form associated to L.¤¼ÔKûÿûÿûÿûÿŒø¼€
__sig__7´ÜŒûÿÿÿŒCRepresentationNumber°§_;DÿÿÿÿÿÿÿÿE0The theta series of the binary quadratic form f.H¢N;F‰ÞÿûÿûÿûÿûÿdÅ èdHèûÿÿÿI6/usr/local/Magma2.8/package/QuadBin/quadratic_orders.m	@W	J6/usr/local/Magma2.8/package/QuadBin/quadratic_orders.m	@W	KDThe order in a quadratic number field of the same discriminant as Q.LL¨ûÿûÿûÿûÿ8(,M
__sig__0ÜôN(4ûÿÿÿdO
QuadraticOrderÈPÿÿÿÿÿÿÿÿ0QZThe regulator of the quadratic order associated to Q, where Q is of positive discriminant.XpR¨VûÿûÿûÿûÿxFP(D˜S
__sig__1¨ÀT(LûÿÿÿU&/usr/local/Magma2.8/package/Lat/auto.m	@W	V*/usr/local/Magma2.8/package/Lat/standard.m	XT	@W	W*/usr/local/Magma2.8/package/Lat/standard.m T	@W	X%The standard lattice given by S and n`YÔûÿûÿûÿûÿ|l\`tZ
__sig__0x[\hûÿÿÿ¼\&/usr/local/Magma2.8/package/Lat/code.m	@W	]&/usr/local/Magma2.8/package/Lat/code.m	@W	^9The lattice constructed from code C according to method SÄÜ_HÔûÿûÿûÿûÿ „txècV&/usr/local/Magma2.8/package/Lat/cons.m	ˆ_	dSThe ambient space of the lattice L, i.e. the tensor product of L with the rationalsdÆþ;eÔtûÿûÿûÿÏœŒf
__sig__0B€Y8gŒ˜ûÿÿÿdh$pThe coordinate space of the lattice L, i.e. the tensor product of the coordinate lattice of L with the rationalsdB€Y8iÔtûÿûÿûÿ¬Œ j
__sig__1C€Y8kŒ¨ûÿÿÿd£ÔÔÔûÿûÿûÿûÿÐS”ŒˆL¥Œûÿÿÿ(¦ The exterior square of lattice L4§ÔÔûÿûÿûÿûÿ¤Œ˜H¨
	__sig__16Ld©Œ ûÿÿÿª!The symmetric square of lattice Lø «ÔÔûÿûÿûÿûÿ´Œ¨@¬
	__sig__17@`­Œ°ûÿÿÿÀ®The dual Lattice of L°
	__sig__18pl
CoordinateSpaceõÂÂ:mÿÿÿÿÿÿÿÿ¤q
__sig__4ýÿpŒÄûÿÿÿwÔûÿûÿûÿûÿDëäŒØsýÿÔÔûÿûÿûÿûÿl{ÔŒÈdt
__sig__3ó-9uŒÐûÿÿÿ–±j9oThe content of the lattice L.P¸ë:nÔûÿûÿûÿûÿÀŒ¼`
__sig__0at€ûÿÿÿb&/usr/local/Magma2.8/package/Lat/cons.m	h”	wÔ¼ûÿûÿûÿûÿä!`!Ø!Hy`!à!ûÿÿÿ$z
SpinorGenus€˜{ÿÿÿÿÿÿÿÿÜ!| The spinor genera in the genus G,D}¼ûÿûÿûÿûÿü!`!ð!ì~
__sig__5ÄÜ`!ø!ûÿÿÿ<€
SpinorGenera|ÿÿÿÿÿÿÿÿô!Š
__sig__7nLoc|ƒ½ûÿûÿûÿûÿûÿ"`!"¼„
__sig__6@è
__sig__8!D!˜!TÔûÿûÿûÿûÿX!Ü L!¬U
__sig__4
$VÜ T!ûÿÿÿÐW(/usr/local/Magma2.8/package/Lat/genera.m(d	 f	X2(/usr/local/Magma2.8/package/Lat/genera.mpd	 f	Y
Representativeð[
IsSpinorÔì\
LocalGeneraÄÜ]Representativesü$<Z
”•–—`
ScaleFactors¼aJordanComponentsôc¼¼ûÿûÿûÿ”!`!ˆ!Td
__sig__0`Øe`!!ûÿÿÿ,!Lÿÿÿÿÿÿÿÿ !MThe deep holes of lattice LNÔûÿûÿûÿûÿ@!Ü 4!òó-9O
__sig__3tPÜ <!ûÿÿÿQ
	DeepHolesTRÿÿÿÿÿÿÿÿ8!S The covering radius of lattice L‚@The coordinate lattice similar to L with Gram matrix scaled by nƒÔÔûÿûÿûÿûÿ$Œ„
__sig__8…Œûÿÿÿ†
ScaledLattice‡ÿÿÿÿÿÿÿÿˆ@The coordinage lattice similar to L with Gram matrix scaled by n‰ÔÔûÿûÿûÿûÿ,Œ Š
__sig__9‹Œ(ûÿÿÿŒThe sum of lattices L1 and L2ÔÔÔûÿûÿûÿûÿÐT<Œ0Ž
	__sig__10Œ8ûÿÿÿ0The orthogonal direct sum of lattices L1 and L2.‘ÔÔÔûÿûÿûÿûÿlLŒ@’
	__sig__11“ŒHûÿÿÿ”0The orthogonal direct sum of lattices L1 and L2.•ÔÔÔûÿûÿûÿûÿ|\ŒP–
	__sig__12—ŒXûÿÿÿÑÿÿÿÿÿÿÿÿ4Ò?The density of L, i.e. the density of the lattice packing of L.$<dÓÔþÿûÿûÿûÿûÿTŒHPÔ
	__sig__25ˆ°ÕŒPûÿÿÿ0Ö
Return s * L,×ÔÔûÿûÿûÿûÿldŒX¼Ø
	__sig__26ÔìÙŒ`ûÿÿÿ|Ú
Return L * s”ÛÔÔûÿûÿûÿûÿ|tŒh€Ü
	__sig__27<¼
CentreDensityÔ½ÿÿÿÿÿÿÿÿä¾The center density of L.Œ¤¿Ôþÿûÿûÿûÿûÿ$ŒøÀ
	__sig__21üÁŒûÿÿÿ`Â
CenterDensity\ÃÿÿÿÿÿÿÿÿüÄ)The centre density of L, in real field K.ÔìÅýÿÔþÿûÿûÿûÿûÿŒÆ
	__sig__22ÇŒûÿÿÿ€Ê
	__sig__23<¯ÔÔûÿûÿûÿûÿOÄŒ¸±ŒÀûÿÿÿ²^The dual quotient L#/L of L, together with the dual L# and the natural epimorphism f:L -> L#/L³Ô“ÔûÿûÿÔŒÈ´
	__sig__19µŒÐûÿÿÿ¶
DualQuotient·ÿÿÿÿÿÿÿÿÌ¸The centre density of L.¹ÔþÿûÿûÿûÿûÿìŒàº
	__sig__20»ŒèûÿÿÿÝŒpûÿÿÿØÞ
Return s * LœßÔÔûÿûÿûÿûÿŒ„Œxœà
	__sig__28|”áŒ€ûÿÿÿÄâ
Return L * sHãÔÔûÿûÿûÿûÿÈ2”Œˆðä
	__sig__29ÈàåŒûÿÿÿ@æ
Return L / s€çÔÔûÿûÿûÿûÿ¬¤Œ˜€è
	__sig__30xéŒ ûÿÿÿXêReturn L / sÜôÈ)The center density of L, in real field K.ÉýÿÔþÿûÿûÿûÿûÿ,Œ ËŒ(ûÿÿÿÌYThe density of L, i.e. the density of the lattice packing of L, returned in real field K.ÍýÿÔþÿûÿûÿûÿûÿL<Œ0Î
	__sig__24ÏŒ8ûÿÿÿ$
MinkowskiSpace„%ÿÿÿÿÿÿÿÿ„ &'/usr/local/Magma2.8/package/Lat/kappa.mø	''/usr/local/Magma2.8/package/Lat/kappa.mø	(The Kappa lattice K_nÀ)Ôûÿûÿûÿûÿ¬ œ   ð*
__sig__0Ôì+œ ¨ ûÿÿÿ,
KappaLattice<-ÿÿÿÿÿÿÿÿ¤ .(/usr/local/Magma2.8/package/Lat/lambda.m	ø	/(/usr/local/Magma2.8/package/Lat/lambda.mX	ø	0The laminated lattice of rank n`Ø
DiagonalizationÔÿÿÿÿÿÿÿÿô2§A lattice with Gram matrix equal to that of L, embedded in an ambient space with a diagonal inner product matrix; the second return value is the transformation matrix.°üÔÔûÿûÿûÿ Œ ¼
	__sig__36”¬Œ ûÿÿÿü0¤The real lattice L given by the image of R under the Minkowski map, with inner product given by the T2-norm on the number field, followed by the isomorphism R -> L.ÔûÿûÿûÿL $ Œ ä»çëÔÔûÿûÿûÿûÿØ2´Œ¨ì
	__sig__31íŒ°ûÿÿÿîTApply Seysen reduction to L, returning a new lattice and the transformation matrix TïÔÔûÿûÿûÿÄŒ¸ð
	__sig__32ñŒÀûÿÿÿòRApply pair-reduction to L, returning a new lattice and the transformation matrix TóÔÔûÿûÿûÿÔŒÈô
	__sig__33õŒÐûÿÿÿþ
	__sig__35itionö&The sequence of orthogonal components.÷ÔûÿûÿûÿûÿäŒØø
	__sig__34ùŒàûÿÿÿúOrthogonalDecompositionûÿÿÿÿÿÿÿÿÜüMThe diagonalization of a symmetric matrix over the rationals or a real field.ýûÿûÿûÿûÿüŒðÿŒøûÿÿÿ1ÔûÿûÿûÿûÿÌ ¼ À L2
__sig__0
Ø
ð
3¼ È ûÿÿÿœ4LaminatedLattice¬Ä5ÿÿÿÿÿÿÿÿÄ 6'/usr/local/Magma2.8/package/Lat/holes.mð‹	7
'/usr/local/Magma2.8/package/Lat/holes.mð‹	8
DirichletVert :DirichletPlanes@Xp;(€The Voronoi cell of lattice L, as the vertices (a sequence of vectors), the set of edges, and the planes (a sequence of vectors) <Ôûÿûÿø Ü ì Ü	__sig__41ÿÿ
"kA lattice with Gram matrix equal to that of L embedded in the Euclidean ambient space over the real field KýÿÔÔûÿûÿûÿûÿ< 4 Œ( 
	__sig__38Œ0 ûÿÿÿ$qA lattice with Gram matrix equal to that of L embedded in the Euclidean ambient space over the default real fieldÔÔûÿûÿûÿûÿD Œ8 
	__sig__40Œ@ ûÿÿÿ	Œ  ûÿÿÿ0¤The real lattice L given by the image of I under the Minkowski map, with inner product given by the T2-norm on the number field, followed by the isomorphism I -> L.ÔûÿûÿûÿÔäT ŒH 
	__sig__42ŒP ûÿÿÿ0¤The real lattice L given by the image of R under the Minkowski map, with inner product given by the T2-norm on the number field, followed by the isomorphism R -> L.Ôûÿûÿûÿt d ŒX 
	__sig__43Œ` ûÿÿÿMinkowskiLatticeÿÿÿÿÿÿÿÿ\ 0¤The real lattice L given by the image of I under the Minkowski map, with inner product given by the T2-norm on the number field, followed by the isomorphism I -> L.Ôûÿûÿûÿ| Œp 
	__sig__44Œx ûÿÿÿ *ŠThe Minkowski vector space V of F, as a real vector space, with inner product given by the T2-norm on F, followed by the embedding F -> V.!úÿÔûÿûÿûÿŒ Œ€ "
	__sig__45#Œˆ ûÿÿÿfHackobjCoerceSymGen¬ÔgÿÿÿÿÿÿÿÿŒ!n
The genus of G$i½½ûÿûÿûÿ¬!`! ! j
__sig__1`k`!¨!ûÿÿÿo¼¼ûÿûÿûÿûÿÌ!Ä!`!¸!|q`!À!ûÿÿÿ¼rThe genus of L.ì$LsÔ¼ûÿûÿûÿûÿP­Ô!`!È!Dt
__sig__34Tu`!Ð!ûÿÿÿèvThe spinor genus of LØx
__sig__4 Ð ü =
__sig__0¬Ô>Ü ô ûÿÿÿ?
VoronoiCell¤
¼
@ÿÿÿÿÿÿÿÿð AThe Voronoi graph of lattice L BÔìÿûÿûÿûÿûÿ!Ü !`C
__sig__1D\DÜ !ûÿÿÿ|E
VoronoiGraph„Fÿÿÿÿÿÿÿÿ!GThe holes of lattice LìHÔûÿûÿûÿûÿ(!Ü !ÜI
__sig__2TJÜ $!ûÿÿÿw	Ôûÿûÿûÿûÿ&ä%Ä$Ø%Lz	RThe symmetric invariant forms of G (the first symmetric form is positive definite)4{	Ôûÿûÿûÿûÿ&ô%Ä$è%H|	
	__sig__13Ld}	Ä$ð%ûÿÿÿ~	'The anti-symmetric invariant forms of G 8X	Ôûÿûÿûÿûÿ,&&Ä$ø%€€	
	__sig__14x˜	Ä$&ûÿÿÿð‚	9n invariant forms of G (either symmetric or antisymmetric¨Àƒ	Ôûÿûÿûÿûÿ&Ä$&l"„	
	__sig__15Ü#$
utiononA	jûÿûÿûÿûÿ¼%%Ä$%,
C	Ä$%ûÿÿÿD
D	NumberOfSymmetricForms\
E	ÿÿÿÿÿÿÿÿ%H	
__sig__2!€!©¼ûÿûÿûÿûÿÐ5¬"`! "H«`!¨"ûÿÿÿ$­ÿÿÿÿÿÿÿÿ¤"®The prime of the genus symbol.¯½ûÿûÿûÿûÿŒõÄ"`!¸"À°
	__sig__14¤¼±`!À"ûÿÿÿÜ²The p-adic determinant of G.l³½ûÿûÿûÿûÿÜ"Ô"`!È"”´
	__sig__15¤¼µ`!Ð"ûÿÿÿ¶The determinant of G.¼·¼ûÿûÿûÿûÿÐ°ä"`!Ø"h"ÿÿÿÿÿÿÿÿ$"½½ûÿûÿûÿûÿL"D"`!8"h‘`!@"ûÿÿÿ’
8P“¼¼ûÿûÿûÿûÿübT"`!H"”
__sig__9tŒ•`!P"ûÿÿÿ¬ž
	__sig__11tive¼–A representative lattice of G.|—¼Ôûÿûÿûÿûÿ,#d"`!X"|˜
	__sig__10tŒ™`!`"ûÿÿÿTšInternalRepresentativeŸ`!x"ûÿÿÿ˜!°!è!lHackobjCoerceSymGenLocìmÿÿÿÿÿÿÿÿ¤!p
__sig__2!H!œ!	Ôûÿûÿûÿûÿ|$$p$H	
__sig__4
Ô
ì
	$x$ûÿÿÿ˜ 	 eThe p-neighbor adjacency matrix with respect to the ordered sequence of representatives of the genus.D!	¼ûÿûÿûÿûÿŒ$$€$ì"	
__sig__5ÄÜ#	$ˆ$ûÿÿÿ<$	%The p-neighbor of L with respect to v<%	ÕÔÔûÿûÿûÿûÿœ$$$&	
__sig__64L'	$˜$ûÿÿÿ<(	
Neighbor h )	ÿÿÿÿÿÿÿÿ”$,	
__sig__7$ $8$	ÿÿÿÿÿÿÿÿ,$	BThe closure of the lattice sequence under the p-neighbor relation.Œ¤	ÔûÿûÿûÿûÿL$$@$	$H$ûÿÿÿD	
NeighborClosure`	ÿÿÿÿÿÿÿÿD$	0The closure of L under the p-neighbour relation.äü	ÔÔûÿûÿûÿûÿd$$X$$	
__sig__3$D	$`$ûÿÿÿŒ	NeighbourClosureT	ÿÿÿÿÿÿÿÿ\$	The immediate p-neighbors of L.h"€"˜"4	Sforms$öAutomorphousClassesd÷ÿÿÿÿÿÿÿÿÌ#ø$sA set of integer representatives of the p-adic automorphous square classes for the lattice L (see Conway & Sloane).ùÔûÿûÿûÿûÿì#|#à#Efè3ú
__sig__4\!û|#è#ûÿÿÿØ
Representatives„ÙÿÿÿÿÿÿÿÿT#Ú$The number of genus representatives.¤Û¼ûÿûÿûÿûÿä„t#`!h#Ü
	__sig__24üÝ`!p#ûÿÿÿ`Þ*/usr/local/Magma2.8/package/Lat/diagonal.mø±	´	ß*/usr/local/Magma2.8/package/Lat/diagonal.m@²	´	à"mTrue if and only if r is coprime to 2 and the determinant and is the norm of an element of the spinor kernel.á¼ûÿûÿûÿûÿŒ#|#€#Tâ
__sig__0`Øã|#ˆ#ûÿÿÿ,!äIsSpinorNorm"˜"°"H#¸
	__sig__16¬Ô¹`!à"ûÿÿÿºThe rank of the genus.ì
»¼ûÿûÿûÿûÿ#ô"`!è"ð¼
	__sig__17Øð½`!ð"ûÿÿÿÄ¾The rank of the genus.ð¿¼ûÿûÿûÿûÿ##`!ø"„À
	__sig__18„œÁ`!#ûÿÿÿ<ÂThe rank of the genus.”Ã½ûÿûÿûÿûÿ”_#`!#€Ä
	__sig__19<Å`!#ûÿÿÿ° ÆThe rank of the genus.h"€"˜"åÿÿÿÿÿÿÿÿ„#æ5A sequence of primes which generate the spinor group.ð
ç¼ûÿûÿûÿûÿ¤#|#˜#ôè
__sig__1Üôé|# #ûÿÿÿÈêSpinorGeneratorsÜôëÿÿÿÿÿÿÿÿœ#ìAA sequence of Dirichlet characters dual to the spinor generators.(Pí¼ûÿûÿûÿûÿ¼#|#°#Hî
__sig__28Xï|#¸#ûÿÿÿìðSpinorCharactersdÜñÿÿÿÿÿÿÿÿ´#6	NendsL#Ç½ûÿûÿûÿûÿ„_$#`!#È
	__sig__20É`! #ûÿÿÿÐ
	__sig__22tiveÊ@A canonical representative for the p-adic genus, in Jordan form.Ë½Ôûÿûÿûÿûÿ4#`!(#Ì
	__sig__21Í`!0#ûÿÿÿÎThe local genus symbols.Ï¼ûÿûÿûÿûÿD#`!8#Ñ`!@#ûÿÿÿÒ
LocalGeneraÓÿÿÿÿÿÿÿÿ<#5	AformsssesÔ1A sequence of representatives for the genus of G.Õ¼ûÿûÿûÿûÿ\#`!P#Ö
	__sig__23×`!X#ûÿÿÿ	+/usr/local/Magma2.8/package/Lat/neighbors.m@¿	ÀÂ		+/usr/local/Magma2.8/package/Lat/neighbors.mˆ¿	ÀÂ		JThe genus of L as isometry class representatives for the p-neighbors of L.Øð	Ôûÿûÿûÿûÿ$$$0	
__sig__0,	$$ûÿÿÿL	SpinorRepresentatives„		ÿÿÿÿÿÿÿÿ$
	JThe genus of L as isometry class representatives for the p-neighbors of L.d|	Ôûÿûÿûÿûÿ4$$($¼	
__sig__1@è	$0$ûÿÿÿè!	GenusRepresentatives#À#ò$qA set of integer representatives of the p-adic automorphous square classes for the genus G (see Conway & Sloane).$ó¼ûÿûÿûÿûÿä#Ô#|#È# ô
__sig__3`õ|#Ð#ûÿÿÿ	
__sig__2vesðü*‡A lattice diagonalized lattice p-adically equivalent to L; if p is 2 then the diagonalized form may have Jordan blocks of dimension 2. Ld|ýÔÔûÿûÿûÿûÿü#|#ð#|þ
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__sig__1$P$h$|Ð1ì1ûÿÿÿÐ&}The absolute norm of x over Q.)~ûÿûÿûÿûÿ2Ð1ô1l/__sig__211D1\1t1›
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__sig__3Üô 82|2ûÿÿÿd¡The ramification index for I4¢ûÿûÿûÿûÿ282„2°£
__sig__4 À¤82Œ2ûÿÿÿP¥The interia degree of IÀØð¦ûÿûÿûÿûÿl‰ 282”2à§
__sig__5X¨82œ2ûÿÿÿ©PTwo generators and g for the ideal I forming a two-element g-normal presentation#Ü#$ªûÿûÿûÿ°282¤2x(«
__sig__6*Ø*ð*¬82¬2ûÿÿÿ-¯
__sig__7/È/0°82¼2ûÿÿÿ¨1ê)/usr/local/Magma2.8/package/Cop/univmap.mXÉ	`Õ	RRootctctë(The universal map from C to S given by Qdìÿ÷ÿeûÿûÿûÿûÿ¸3¨3¬3í
__sig__0dB€Y8î¨3´3ûÿÿÿï
UniversalMapê.:ðÿÿÿÿÿÿÿÿ°3ñ,/usr/local/Magma2.8/package/AlgFP/groebner.mË	`Õ	ò,/usr/local/Magma2.8/package/AlgFP/groebner.mË	`Õ	ól‘Input: a finitely generated free group F and two finitely generated subgroups H1 and H2; Output: a coset graph CG for the intersection of H1 and H2. This graph is the component of the graph CG1xCG2 which includes the vertex (1,1) (where CG1, CG2 are the coset graphs of H1, H2 respectively). The vertex labels of CG are minimal with respect to the Shortlex ordering with regard to the generators of F dC€Y8ô"""ÿÿûÿûÿûÿûÿØ3È3Ì3õ
__sig__0EX;öÈ3Ô3ûÿÿÿd&|An isomorphism from the i-th perfect group G in the database as a fp group onto a representation of G as a permutation group(
__sig__5„œ4t4ûÿÿÿÄ*‡An isomorphism from the i-th perfect group G of order o in the database as a fp group onto a representation of G as a permutation group`x 5"æûÿûÿ4ˆ44|4@!
__sig__6h€"4„4ûÿÿÿð#$sAn isomorphism from the perfect group G named 'top' as a fp group onto a representation of G as a permutation groupX$5"æûÿûÿ 4˜44Œ4ì!%
__sig__7#d#”#&4”4ûÿÿÿ%'(„An isomorphism from the perfect group G which is 'top'#prime<exp, n> as a fp group onto a representation of G as a permutation group--(5"æûÿûÿ°4¨44œ4ø3÷CosetGraphIntersect)øÿÿÿÿÿÿÿÿÐ3ù.šInput: a finitely generated free group F and two finitely generated subgroups H1 and H2; Output: the subgroup of F which is the intersection of H1 and H2 3ú""""ûÿûÿûÿûÿð3È3ä3hTÉ:û
__sig__14-üÈ3ì3ûÿÿÿý
FGIntersect„0þÿÿÿÿÿÿÿÿè3__sig__2Æûÿûÿûÿûÿ• 3823È823ûÿÿÿÙ
tGroupeÉ7The representative of I in the basis of the class groupÊûÿûÿûÿûÿ0382$3Ë
	__sig__14Ì82,3ûÿÿÿÍ4The unit group of the maximal order of Q (i.e. of Z)Î“ûÿûÿûÿPF@38243Ï
	__sig__15Ð82<3ûÿÿÿÒ,–Given a prime ideal in some order O, return a function that decides whether its argument (an element of O) is a square in the completion of O at pid. 82Óä†P3ûÿÿÿÆ!MakeIsSquareÖ
e unit groÕ“ûÿûÿûÿ\3ä†X3Ø
__sig__3`382×ä†`3ûÿÿÿÔ__sig__4ÿÿÿÚûÿûÿûÿûÿx3p382d3Û
	__sig__18Ü82l3ûÿÿÿÝ
Þûÿûÿûÿûÿ|‰€382t3ß
	__sig__19à82|3ûÿÿÿá
âûÿûÿûÿ˜3382„3ã
	__sig__20ä82Œ3ûÿÿÿåNThe signature (number of real embeddings and pairs of complex embeddings) of Qæûÿûÿûÿ 382”3ç
	__sig__21è82œ3ûÿÿÿé)/usr/local/Magma2.8/package/Cop/univmap.mÞ	h	B./usr/local/Magma2.8/package/GrpFP/solquot/sq.m	¨ê	D"iCompute a soluble quotient of F with order n. If n is 0, then compute largest possible soluble quotient. E55ûÿÿÿH.šCompute a soluble quotient of F. P is a set of primes and the algorithm calculates the largest quotient such that the order has prime divisors only in P. :G"ûÿ,5$55 5':J
__sig__6ØI5(5ûÿÿÿLHÿCompute a soluble quotient of F. S is a sequence of tuples <p, e>, with p a prime or 0 and e a non-negative integer. The order of the quotient will be a divisor of &* [ p^e : <p, e> in S] if all p's and e's are positive. See handbook for further details. 5,5äN
__sig__7dM585ûÿÿÿP3/usr/local/Magma2.8/package/GrpFP/solquot/sq_proc.m@â	 ä	Oâ3/usr/local/Magma2.8/package/GrpFP/solquot/sq_proc.mâ	 ä	ZModi¥XSeriesand eYDatabook fo]
SplitExt¦§[Limits^NonsplitExt¦§¨©\ModulCandidates__sig__3`
Collect¥¦§S0Compute largest possible soluble quotient of F. ¤T"ûÿ5X55L5U
__sig__4V5T5ûÿÿÿ<C"ûÿ5555K"ûÿ45505d_RelatP¸ë:Q£¤¥¦§¨©ª«¬­a"gInitialize a SQProc for a finitely presented group F without any information about the relevant primes.d‹ñ'7b"sûÿûÿûÿûÿ 55<5„5dc
__sig__0ôî6d<5Œ5ûÿÿÿp[;”sssûÿûÿX6<5L6H•
	__sig__10Ô
ì
–<5T6ûÿÿÿ˜—IntersectKernels¨À˜ÿÿÿÿÿÿÿÿP6™6»Compose the lifts SQ1 and SQ2 of SQP to a new bigger quotient. If the optional parameter Check is set to true, it will be tested whether the intersection of the kernels has maximal index.Tlšssssûÿûÿûÿp6<5d6”›
	__sig__11¤¼œ<5l6ûÿÿÿComposeQuotients¤¼žÿÿÿÿÿÿÿÿh6·
	__sig__17cessh"Ÿ&xStart the soluble quotient algorithm for a finitely presented group F without any information about the relevant primes.(Œ(¤( "sûÿûÿûÿ˜6ˆ6<5|6¼.¡
	__sig__12Ü01¢<5„6ûÿÿÿÌ4…
ReplacePrimesÔ†ÿÿÿÿÿÿÿÿ6‡.Check the correctness of the soluble quotient.hˆsûÿûÿûÿûÿ(6<56ð‰
__sig__8\tŠ<5$6ûÿÿÿÀ‹
SQ_check¤¼Œÿÿÿÿÿÿÿÿ 66ºChecks whether two soluble quotients have the same kernel. If not and Construct is true, then a bigger quotient will be constructed, where the kernel is the intersection of both kernels.tŒŽsssûÿûÿûÿ@6<546è
__sig__9!D!˜!<5<6ûÿÿÿ#‘EquivalentQuotientsø$%’ÿÿÿÿÿÿÿÿ86“6¾Construct a bigger soluble quotient by intersecting the kernels of the given quotient. The return values are a new soluble quotient process and maps from the new to the given soluble groups.ì5e

Initializeœ¼fÿÿÿÿÿÿÿÿˆ5g fInitialize a SQProc for a given quotient epi:F -> G without any information about the relevant primes.Thsûÿûÿûÿûÿ°5¨5<5œ5i
__sig__1 !,!j<5¤5ûÿÿÿH#kKInitialize a SQProc for a finitely presented group F with expected order n.X'x'l"sûÿûÿûÿûÿÀ5¸5<5¬5ì,m
__sig__2./l/n<5´5ûÿÿÿt1ÙModulesrs)
__sig__8*4¤4ûÿÿÿ+*ŠA homomorphism from G onto a permutation group P defined by R (which should be an element of the sequence returned by Group(D[pfgps], ...),"æûÿûÿûÿ¸44¬4-
__sig__9.4´4ûÿÿÿ/"hThe number of available representations of the i-th perfect group in the database as a permutation group05ûÿûÿûÿûÿØ4È44¼41
	__sig__1024Ä4ûÿÿÿ3NumberOfRepresentations4ÿÿÿÿÿÿÿÿÀ45$sThe number of available representations of the i-th perfect group of order o in the database as a permutation group65ûÿûÿûÿûÿè4à44Ô47
	__sig__1184Ü4ûÿÿÿ9$oThe number of available representations of the perfect group named 'top' in the database as a permutation group:5ûÿûÿûÿûÿø4ð44ä4;
	__sig__12<4ì4ûÿÿÿ="gThe number of available representations of the perfect group 'top'#prime<exp, n> as a permutation group>5ûÿûÿûÿûÿ54ô4?
	__sig__13@4ü4ûÿÿÿA./usr/local/Magma2.8/package/GrpFP/solquot/sq.m	h	Èaûÿûÿûÿûÿ(???É
__sig__0Ê?$?ûÿÿÿ@-Replace x with (x * y)Ô.ÿÿûÿûÿûÿûÿûÿØÀ@@@´@$/
__sig__7€˜0@@¼@ûÿÿÿÀ1Replace x with (x / y)ð2ÿÿûÿûÿûÿûÿûÿÐ@@@Ä@,3
__sig__84@@Ì@ûÿÿÿT5Replace x with (x ^ y)œ6ÿÿûÿûÿûÿûÿûÿPtà@@@Ô@Ü7
__sig__9Ôì8@@Ü@ûÿÿÿL9Replace x with (x div y)°:ÿÿûÿûÿûÿûÿûÿð@@@ä@H#;
	__sig__10P$h$<@@ì@ûÿÿÿÐ&=Replace x with (x mod y)(¼()>ÿÿûÿûÿûÿûÿûÿA@@ô@l/?
	__sig__111D1@@@ü@ûÿÿÿì5AReplace x with (x and y)78l8BÿÿûÿûÿûÿûÿûÿA@@Aü;C
	__sig__12Ô<ì<D@@Aûÿÿÿ¤>Eand:=¤?äÿÿÿÿÿÿÿÿ€?å&The graph of the the Hadamard matrix Hì
æïÿaûÿûÿûÿûÿ ??”?ðç
__sig__5Øðè?œ?ûÿÿÿÄé
HadamardGraphÀêÿÿÿÿÿÿÿÿ˜?ëThe Clebsch graphl„ìaûÿûÿûÿûÿ¸??¬?¼í
__sig__6ÄÜî?´?ûÿÿÿï
ClebschGraphLðÿÿÿÿÿÿÿÿ°?ñThe Shrikhande graph!,!òaûÿûÿûÿûÿÐ??Ä?%ó
__sig__7&è&'ô?Ì?ûÿÿÿ)õ
ShrikhandeGraphü+öÿÿÿÿÿÿÿÿÈ?÷&The sequence of the three Chang graphs¼3øûÿûÿûÿûÿè??Ü?œ9ù
__sig__8:´:Ì:ú?ä?ûÿÿÿ,<û
ChangGraphs==ýThe Gewirtz graph¼>Ü>,?D?Ë
OddGraphŒ>Ï__sig__14Öÿ“ûÿûÿûÿ°FØD8DÌD$7=The order of the group of divisor classes of degree zero of FP8ÖÿûÿûÿûÿûÿàFèD8DÜD„9
__sig__3˜:8DäDûÿÿÿ°;*‹The group of global units of F, i.e. the multiplicative group of the exact constant field, as an Abelian group together with the map into Fœ<Öÿ“ûÿûÿûÿøD8DìDÐ=
__sig__4ä>8DôDûÿÿÿü?
GlobalUnitGroup¬	@ÿÿÿÿÿÿÿÿðDA+Whether a is a global unit, i.e. a constantð	BáûÿûÿûÿûÿE8DE$
C
__sig__5
8
D8DEûÿÿÿP
E
IsGlobalUnitÜFÿÿÿÿÿÿÿÿEO
__sig__7reimage$GDTrue and the preimage of a in the global unit group, false otherwiseh5I
__sig__65|5‚sûÿûÿûÿûÿûÿ6<56ƒ
__sig__7„<56ûÿÿÿ¹NonsplitCollector¬ÔºÿÿÿÿÿÿÿÿØ6»,”Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails.ð¼sûÿûÿûÿûÿûÿ7ø6<5ì6,½
	__sig__18¾<5ô6ûÿÿÿT¿6¸Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails. ws defines the weights of a series.tŒÀsûÿûÿûÿûÿûÿ77<5ü6èÁ
	__sig__19D!˜!Â<57ûÿÿÿ#ÃBêSet up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails. ws defines the weights of a series. epi is an epimorphism onto another soluble group.œ0´0Å
	__sig__20l2¼3Æ<57ûÿÿÿŒ6£SolubleQuotientProcess$¤ÿÿÿÿÿÿÿÿ€6¥$rStart the soluble quotient algorithm for a finitely presented group F with expected order of the quotient being n.8P¦"sûÿûÿûÿ¨6 6<5”6§
	__sig__13tŒ¨<5œ6ûÿÿÿ¬©*‰Start the soluble quotient algorithm for a finitely presented group F with expected order of the quotient being the factorized integer f.t”ª²"sûÿûÿûÿ¸6°6<5¤6€«
	__sig__14<¬<5¬6ûÿÿÿ° ­(‚Start the soluble quotient algorithm for a finitely presented group F with relevant primes restricted by the sequence of tuples f.è&'®"sûÿûÿûÿÈ6À6<5´6¬,¯
	__sig__15d.¼.°<5¼6ûÿÿÿD1³
	__sig__164Ì4sûÿûÿûÿûÿP8H8<5<8L<5D8ûÿÿÿ(@âCalculates the nonsplit extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).xsûÿûÿûÿûÿX8<5L8@
	__sig__37h€<5T8ûÿÿÿðXDelete the k-th split solution space of the i-th p-module as the actual solution space. 8Lsûÿûÿûÿûÿûÿh8<5\8!
	__sig__384"l"<5d8ûÿÿÿ<$DeleteSplitSolutionspace&Œ&Ô&ÿÿÿÿÿÿÿÿ`8XDelete the k-th split solution space of the i-th p-module as the actual solution space. 0 0¸0sûÿûÿûÿûÿûÿ€8<5t87¨7À7±$sStart the soluble quotient algorithm for a finitely presented group F with relevant primes restricted by the set s.²"sûÿûÿûÿÐ6<5Ä6´<5Ì6ûÿÿÿµ"kSet up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p.¶sûÿûÿûÿûÿûÿð6à6<5Ô6¸<5Ü6ûÿÿÿÚÿÿÿÿÿÿÿÿX7Û7Calculates the modules for all (known) relevant primes.ì
$Üsûÿûÿûÿûÿx7<5l7 Ý
	__sig__25`Þ<5t7ûÿÿÿQxor:=tionßT4Append/Find the module M in the list of modules in SQP. The first return value is the index of the module in the char p list in SQP. The second return value is false iff the index belongs to an isomophic module. (It might happen that the isomorphism is the identity; relevant is the internal data structure.)àsûÿûÿûÿˆ7<5|7° á
	__sig__26""â<5„7ûÿÿÿ$ã
AppendModule%@%äÿÿÿÿÿÿÿÿ€7å
	internal Ô*ì*æsûÿûÿûÿ 7<5”7´0ç
	__sig__27l2¼3è<5œ7ûÿÿÿŒ6ä6,7Äsûÿûÿûÿûÿûÿ7<57Ç$qSet up the collector for a standard split extension of SQP with algebra RG, R the prime ring in characteristic p.Èsûÿûÿûÿûÿûÿ87(7<57É
	__sig__21Ê<5$7ûÿÿÿË
SplitCollectorÌÿÿÿÿÿÿÿÿ 7Í,•Setup the collector for a standard ssplit extension of SQP with algebra RG, R the prime ring in characteristic p. ws defines the weights of a series.ÎsûÿûÿûÿûÿûÿH7@7<547Ï
	__sig__22Ð<5<7ûÿÿÿÑ8ÁSet up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. ws defines the weights of a series. epi is an epimorphism onto another soluble group.ÒsûÿûÿûÿûÿûÿP7<5D7Ó
	__sig__23Ô<5L7ûÿÿÿÕ6¹Calculates the G-modules in characteristic p with respect to the given options. The return value is 0 iff no such module exists, otherwise it is the index of the list of modules in SQP.Ösûÿûÿûÿûÿp7`7<5T7×
	__sig__24Ø<5\7ûÿÿÿNonsplitExtensionSpaceÔÿÿÿÿÿÿÿÿ8<ÑCalculates the nonsplit extension lift for the modules in the list. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).0Åsûÿûÿûÿûÿ08(8<58¬	
	__sig__34œ¼
<5$8ûÿÿÿL>ÝCalculates the nonsplit extension lift for the modules in the l-th list in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).sûÿûÿûÿûÿ@888<5,8À#
	__sig__35ø$%<548ûÿÿÿx'<ÒCalculates the split extension lift for the p-modules stored in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).6,6
	__sig__36¤7¼7é
SolutionSpaceêÿÿÿÿÿÿÿÿ˜7ë4³Calculates the split extension lift for the module M. The return is -1 iff the solution space does not exist in general,otherwise it is the Fq-dimension of the space (could be 0).ìsûÿûÿûÿûÿÈ7¸7<5¬7í
	__sig__28î<5´7ûÿÿÿïSplitExtensionSpaceðÿÿÿÿÿÿÿÿ°7ñ:ÌCalculates the split extension lift for the modules in the list. Returned is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).òÅsûÿûÿûÿûÿØ7Ð7<5Ä7ó
	__sig__29ô<5Ì7ûÿÿÿõ>ÙCalculates the split extension lift for the modules in the l-th list in SQP. The return is a sequence of: -1 iff the solution space does not exist in general,otherwise it is the Fq-dimension of the space (could be 0).ösûÿûÿûÿûÿè7à7<5Ô7÷
	__sig__30ø<5Ü7ûÿÿÿù<ÒCalculates the split extension lift for the p-modules stored in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).úsûÿûÿûÿûÿø7ð7<5ä7û
	__sig__31ü<5ì7ûÿÿÿý@ßCalculates the split extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).þsûÿûÿûÿûÿ8<5ô7ÿ
	__sig__32<5ü7ûÿÿÿ%
	__sig__40tionspace6·Calculates the nonsplit extension lift for the module M. The return is -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).sûÿûÿûÿûÿ 88<58
	__sig__33<58ûÿÿÿ9LiftSplitExtensionRowÔ:ÿÿÿÿÿÿÿÿØ8;\Build the split extensions of SQP for the list of p-modules for their actual solution space.<ssûÿûÿûÿ9ø8<5ì8Ä=
	__sig__45¨À><5ô8ûÿÿÿì?QBuild the split extension of the list of modules for their actual solution space.Ôì@ssûÿûÿûÿ9<5ü8A
	__sig__46B<59ûÿÿÿ€C&~Build the non-split extension of the i-th p-module for the k-th solution space. A set of linear combinations can be specified.H#Dssûÿûÿûÿûÿ(99<59x'E
	__sig__47¼()F<59ûÿÿÿ,,GLiftNonsplitExtension¼.Hÿÿÿÿÿÿÿÿ9K
	__sig__486,6L<5,9ûÿÿÿl8„8œ8!DeleteNonsplitSolutionspace<t"ÿÿÿÿÿÿÿÿx8#&zBuild the split extension of the i-th p-module for the k-th solution space. A set of linear combinations can be specified.¨À$ssûÿûÿûÿûÿ¨8˜8<5Œ8ð&<5”8ûÿÿÿ'LiftSplitExtension4T(ÿÿÿÿÿÿÿÿ8)XBuild the split extension of the l-th list of p-modules for their actual solution space.¤¼*ssûÿûÿûÿ¸8°8<5¤8<+
	__sig__41 h ,<5¬8ûÿÿÿ"-SBuild the split extension of the list of p-modules for their actual solution space.ˆ&Ð&.ssûÿûÿûÿÈ8À8<5´8,,/
	__sig__42.,.0<5¼8ûÿÿÿ11QBuild the split extension of the list of modules for their actual solution space.8l8
	__sig__39 <5|8ûÿÿÿI\Build the non-split extension of the l-th list of p-modules for their actual solution space.	Jssûÿûÿûÿ8909<5$9dMWBuild the non-split extension of the list of p-modules for their actual solution space.ÈìNssûÿûÿûÿH9@9<549€O
	__sig__49pˆP<5<9ûÿÿÿðQUBuild the non-split extension of the list of modules for their actual solution space.HRssûÿûÿûÿP9<5D9S
	__sig__50ôT<5L9ûÿÿÿœ!U"iBuild the non-split extensions of SQP for the p-modules in the l-th list for their actual solution space.ì&'Vssûÿûÿûÿp9`9<5T9°,W
	__sig__51h.À.X<5\9ûÿÿÿH1YLiftNonsplitExtensionRow5606]
	__sig__52¨7À72ssûÿûÿûÿÐ8<5Ä83
	__sig__434<5Ì8ûÿÿÿe
	__sig__54ionRow5 eBuild the split extensions of SQP for the p-modules in the l-th list for their actual solution space.6ssûÿûÿûÿð8à8<5Ô87
	__sig__448<5Ü8ûÿÿÿq
PrintQuotient„rÿÿÿÿÿÿÿÿ¸9s
Print functionì
tsûÿûÿûÿûÿûÿØ9<5Ì9ðu
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PrintSeries¨ÀxÿÿÿÿÿÿÿÿÐ9y
Print functionTzsûÿûÿûÿûÿûÿð9<5ä9|{
	__sig__58t”|<5ì9ûÿÿÿì}
PrintPrimesô~ÿÿÿÿÿÿÿÿè9
Print function° €sûÿûÿûÿûÿûÿ::<5ü9h$
	__sig__59H&`&‚<5:ûÿÿÿt(ƒ
PrintCollectorì*„ÿÿÿÿÿÿÿÿ:…
Print function´0†sûÿûÿûÿûÿûÿ :<5:Œ7‡
	__sig__60d9œ9ZÿÿÿÿÿÿÿÿX9[ `Build the non-split extensions of SQP for the list of p-modules for their actual solution space.°Ð\ssûÿûÿûÿ€9x9<5l9°^<5t9ûÿÿÿ0_UBuild the non-split extension of the list of modules for their actual solution space.Ü`ssûÿûÿûÿˆ9<5|9a
	__sig__53$b<5„9ûÿÿÿ|c<Print function for SQProc. Print prime specific information.Tdsûÿûÿûÿûÿûÿ¨9˜9<5Œ9è!f<5”9ûÿÿÿÀ#g
PrintProcess$%hÿÿÿÿÿÿÿÿ9i"General print function for SQProc.ä+ü+jsûÿûÿûÿûÿûÿ°9<5¤9D1k
	__sig__554Ì4l<5¬9ûÿÿÿŒ7Ç/Return the list of all p-modules stored in SQP.¨tÈsÅûÿûÿûÿûÿ8;(;<5;lÊ<5$;ûÿÿÿ”Ë

GetModules|”Ìÿÿÿÿÿÿÿÿ ;Í0Return the l-th list of p-modules stored in SQP.¨ÀÎsÅûÿûÿûÿûÿ@;<54;pÏ
	__sig__73¨ÀÐ<5<;ûÿÿÿ(ç
	__sig__78ectorHÑ>Delete the collector for split extensions in characteristic p.ìÒsûÿûÿûÿûÿûÿ`;P;<5D;œ!Ó
	__sig__74´"L#Ô<5L;ûÿÿÿ¼$ÕDeleteSplitCollector&'ÖÿÿÿÿÿÿÿÿH;×+Delete all collectors for split extensions..0.Øsûÿûÿûÿûÿûÿh;<5\;ð5Ù
	__sig__75h77Ú<5d;ûÿÿÿà9'¹m
Print functionnsûÿûÿûÿûÿûÿÀ9<5´9o
	__sig__56p<5¼9ûÿÿÿˆ<5:ûÿÿÿÔ‰
Print function˜Šsûÿûÿûÿûÿûÿ0:<5$:˜‹
	__sig__61xŒ<5,:ûÿÿÿÀŸ
Print functionˆ sûÿûÿûÿûÿûÿˆ:<5|:ð
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	__sig__65l¢<5„:ûÿÿÿ”£7Returns the soluble group and the epimorphism from SQP.0H¤sûÿûÿûÿ˜:<5Œ:ð¥
	__sig__66Èà¦<5”:ûÿÿÿ@§
GetQuotienth€¨ÿÿÿÿÿÿÿÿ:©PReturn the relevant primes stored in SQP and a flag indicating the completeness.d„ªsûÿûÿûÿ°:<5¤:0!«
	__sig__67„"œ"¬<5¬:ûÿÿÿl$­
	GetPrimesL&d&®ÿÿÿÿÿÿÿÿ¨:¯GReturns the parent of SQP, i.e. the soluble Quotient which lifts to SQPÀ./p/°ssûÿûÿûÿûÿÈ:<5¼:`6±
	__sig__688p8²<5Ä:ûÿÿÿP:

PrintRelat ÈŽÿÿÿÿÿÿÿÿ(:
Print functionPsûÿûÿûÿûÿûÿX:H:<5<:P‘
	__sig__62`€’<5D:ûÿÿÿà“
PrintModulesð”ÿÿÿÿÿÿÿÿ@:•
Print function–sûÿûÿûÿûÿûÿ`:<5T:L#—
	__sig__63T$l$˜<5\:ûÿÿÿÔ&™
Print functionx(šsûÿûÿûÿûÿûÿ€:p:<5d:0.›
	__sig__64 0¸0œ<5l:ûÿÿÿø3
PrintExtensions6žÿÿÿÿÿÿÿÿh:³
	GetParent\„´ÿÿÿÿÿÿÿÿÀ:µ;Returns a list of soluble quotients which are lifts of SQP.€˜¶sÅûÿûÿûÿûÿà:<5Ô:`·
	__sig__69œÄ¸<5Ü:ûÿÿÿðÛBDelete the collector for non-split extensions in characteristic p.ôÜsûÿûÿûÿûÿûÿˆ;x;<5l;ÔÝ
	__sig__76¬ÄÞ<5t;ûÿÿÿôßDeleteNonsplitCollectorÈàÿÿÿÿÿÿÿÿp;á/Delete all collectors for non-split extensions. 8Xâsûÿûÿûÿûÿûÿ;<5„;€ã
	__sig__77x˜ä<5Œ;ûÿÿÿðå*Delete the collectors in characteristic p.Xæsûÿûÿûÿûÿûÿ°; ;<5”;ì!è<5œ;ûÿÿÿÄ#é
DeleteCollector%êÿÿÿÿÿÿÿÿ˜;ë!Delete all the collectors of SQP.è+,ìsûÿûÿûÿûÿûÿ¸;<5¬;H1í
	__sig__794Ð4î<5´;ûÿÿÿ7ï-Delete the soluble quotient process variable. :¹
GetChildren8PºÿÿÿÿÿÿÿÿØ:»9Returns the i-th soluble quotient which is a lift of SQP. À¼sÅûÿûÿûÿûÿø:<5ì:½
	__sig__70Ìì¾<5ô:ûÿÿÿÜ¿
GetChild Ô !Àÿÿÿÿÿÿÿÿð:Á'Return the i-th p-module stored in SQP.t%L&d&Âszûÿûÿûÿûÿ;<5;,Ã
	__sig__71è-.Ä<5;ûÿÿÿ¸0ssûÿûÿûÿûÿ<<5<HKeepGeneratorOrder$ÿÿÿÿÿÿÿÿ<*ŒReturn the sequence useful for the Collector setup. It determines that the conjugation action of the pc generators in SQP is kept unchanged.ðssûÿûÿûÿûÿ(<<5<„	
	__sig__84„œ
<5$<ûÿÿÿ<KeepGeneratorActiont”ÿÿÿÿÿÿÿÿ <(‚Return the sequence useful for the Collector setup. It determines that the order of the pc generators in SQRSQP is kept unchanged.°!è!ssûÿûÿûÿûÿ@<<54<p%
	__sig__85è'(<5<<ûÿÿÿ¼*
KeepElementary¬,ÿÿÿÿÿÿÿÿ8<,Return the sequence useful for the Collector setup. It determines that the conjugation action of the pc generators in SQR SQP is kept unchanged.:4:L:½
ction,;Å
	GetModule4Ì4Æÿÿÿÿÿÿÿÿ;É
	__sig__72ô9:ssûÿûÿûÿûÿX<<5L<¤
	__sig__86|¬<5T<ûÿÿÿd
KeepAbelian°ÐÿÿÿÿÿÿÿÿP<0 Return the sequence useful for the Collector setup. It determines that the conjugation action of the pc generators of SQP on those of SQR SQP is kept unchanged.”¬ssûÿûÿûÿûÿp<<5d<¼
	__sig__87Ôì<5l<ûÿÿÿD
KeepGroupActionTÿÿÿÿÿÿÿÿh<$qReturn the sequence useful for the Collector setup. It determines that the presentation of SQP is kept unchanged.¨#À# ssûÿûÿûÿûÿˆ<<5|<D(!
	__sig__88œ*¼*"<5„<ûÿÿÿì,#
	KeepSplit/l/$ÿÿÿÿÿÿÿÿ€<-
	__sig__90ian\6&ssûÿûÿûÿûÿ <<5”<Ì:Kor:=hphðsûÿûÿûÿûÿûÿÈ;<5¼;¬ñ
	__sig__80$ò<5Ä;ûÿÿÿÐó
DeleteProcessÀôÿÿÿÿÿÿÿÿÀ;õ/Delete the Process and all its child processes.ÀØðösûÿûÿûÿûÿûÿà;<5Ô;„÷
	__sig__81„œø<5Ü;ûÿÿÿ<ùDeleteProcessDownt”úÿÿÿÿÿÿÿÿØ;
	__sig__83ete€û>Delete all soluble quotient processes which are related to SQP° üsûÿûÿûÿûÿûÿø;<5ì;h$ý
	__sig__82H&`&þ<5ô;ûÿÿÿt(ÿDeleteProcessComplete´+ÿÿÿÿÿÿÿÿð;(Return the sequence useful for the Collector setup. It determines that the order of the pc generators in SQP is kept unchanged.l8„8œ8<5<ûÿÿÿœ:M<ÕDetermine the maximal abelian module which lifts to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.Ä]
	__sig__99ectionQTDetermine the maxinmal p-elementary abelian module which lifts to a bigger quotient.0Rssûÿûÿûÿ`=P=<5D=ÀS
	__sig__97ØðT<5L=ûÿÿÿ€UElementaryAbelianSectionøVÿÿÿÿÿÿÿÿH=W@àDetermine the maximal elementary abelian module which lifts to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.%L&d&Xssûÿûÿûÿh=<5\=,Y
	__sig__98è-.Z<5d=ûÿÿÿ¸0[IDetermine the maximal p-group with a splitting lift to a bigger quotient.h77\ssûÿûÿûÿˆ=x=<5l=€;^<5t=ûÿÿÿÀ<)KeepElementaryAbelianÔ*ÿÿÿÿÿÿÿÿ˜<+.˜Return the sequence useful for the Collector setup. It determines that the presentation of SQP and the conjugation action in SQR SQP are kept unchanged.`,ssûÿûÿûÿûÿ¸<<5¬<\.<5´<ûÿÿÿ|/KeepSplitAbelianœ¼0ÿÿÿÿÿÿÿÿ°<9
	__sig__92yAbelian1(Return the sequence useful for the Collector setup. It determines that the presentation of SQP and of SQR SQP are kept unchanged.¤¼2ssûÿûÿûÿûÿÐ<<5Ä<h"3
	__sig__91Ø#$4<5Ì<ûÿÿÿp%5KeepSplitElementaryAbelian\(t(6ÿÿÿÿÿÿÿÿÈ<7MReturn a sequence of pairs [j,i] s.t. the order of both G.j and G.i is not p.t18sûÿûÿûÿûÿè<<5Ü<œ8:<5ä<ûÿÿÿœ:;
KeepPrimePowerÌ;¹
t<Œ<%.žReturn the sequence useful for the Collector setup. It determines that the order and the conjugation action of the pc generators in SQR SQP is kept unchanged.'
	__sig__89(<5œ<ûÿÿÿ<ÿÿÿÿÿÿÿÿà<=*ŒReturn a sequence of pairs [j,i] correspnding to the weight condition for the p-group. The weights are defined by the sequence of quotients.>ûÿûÿûÿûÿ=<5ô<Ä?
	__sig__93¨À@<5ü<ûÿÿÿì{
__sig__105ianSectionØo^Determine the maxinmal p-elementary abelian module with a splitting lift to a bigger quotient.¨pssûÿûÿûÿØ=È=<5¼=”q

__sig__103r<5Ä=ûÿÿÿdsSplitElementaryAbelianSectionÀtÿÿÿÿÿÿÿÿÀ=uBéDetermine the maximal elementary abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.Xvssûÿûÿûÿà=<5Ô=ì!w

__sig__104d#”#x<5Ü=ûÿÿÿ%yMDetermine the maximal p-group with a non splitting lift to a bigger quotient.À*zssûÿûÿûÿ>ð=<5ä=p0|<5ì=ûÿÿÿÀ3}
NonsplitSection`6~ÿÿÿÿÿÿÿÿè=€ssûÿûÿûÿ><5ü=<®®AKeepPGroupWeightspˆBÿÿÿÿÿÿÿÿø<C5Check the PC presentation of G for the Hall property.àDûÿûÿ=<5=ìE
	__sig__94´F<5=ûÿÿÿ!G
IsPureOrder4"l"Hÿÿÿÿÿÿÿÿ=IHDetermine the maximal p-abelian module which lifts to a bigger quotient.)`*x*Jssûÿûÿûÿ8=0=<5$=0K
	__sig__95`1x1L<5,=ûÿÿÿ06Nssûÿûÿûÿ@=<54= :O
	__sig__96¨;Ð;P<5<=ûÿÿÿð<_
SplitSection„`ÿÿÿÿÿÿÿÿp=a@àDetermine the maximal nilpotent group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.H`bssûÿûÿûÿ=<5„=Üc

__sig__100äüd<5Œ=ûÿÿÿ|eRDetermine the maximal p-abelian module with a splitting lift to a bigger quotient.¤¼fssûÿûÿûÿ°= =<5”=<g

__sig__101 h h<5œ=ûÿÿÿ"ŽssûÿûÿûÿP>@><54>L<5<>ûÿÿÿ(‘ NonsplitElementaryAbelianSection4’ÿÿÿÿÿÿÿÿ8>“BìDetermine the maximal elementary abelian group with a nonsplitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.P”ssûÿûÿûÿX><5L>H•

__sig__1108X–<5T>ûÿÿÿì—CDetermine a lift with a p-group, given by its lower central series.!0!˜ssûÿûÿûÿh><5\>%š<5d>ûÿÿÿ|'›
PGroupSection)œÿÿÿÿÿÿÿÿ`>P Determine the maximal nilpotent group with a lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account. Steps puts a limit on the weights of the p-groups of the nilpotent group.<¨<À<³
¨=Ð=iSplitAbelianSectionT$l$jÿÿÿÿÿÿÿÿ˜=k>ÞDetermine the maximal abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.`6lssûÿûÿûÿ¸=<5¬=Ð:m

__sig__102è;<n<5´=ûÿÿÿ =X=€=BéDetermine the maximal elementary abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.8P

__sig__106ˆ°‚<5>ûÿÿÿ0

__sig__109ion,ƒVDetermine the maximal p-abelian module with a non splitting lift to a bigger quotient.ü„ssûÿûÿûÿ(>><5>$…

__sig__107$D†<5>ûÿÿÿŒ‡NonsplitAbelianSectionTˆÿÿÿÿÿÿÿÿ>‰BéDetermine the maximal elementary abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.,,,Šssûÿûÿûÿ0><5$>t1‹

__sig__108”5ì5Œ<5,>ûÿÿÿ¼7™
__sig__111belianSection: ]Determine the maximal p-elementary abelian module with a splitting lift to a bigger quotient.|=¤=Ì=°ÿÿÿÿÿÿÿÿ°>±*/usr/local/Magma2.8/package/AlgChtr/pgps.mð¨	`®	²*/usr/local/Magma2.8/package/AlgChtr/pgps.m8©	`®	´ûÿûÿûÿûÿè>Ø>È>Ì>µ
__sig__00¶È>Ô>ûÿÿÿ\·CharacterDegreesd|¸ÿÿÿÿÿÿÿÿÐ>º“ûÿûÿûÿûÿø>ð>È>ä>\»
__sig__1d|¼È>ì>ûÿÿÿÜ¾æûÿûÿûÿûÿ??È>ô>h ¿
__sig__2!°!è!ÀÈ>ü>ûÿÿÿÀ#Á&|The sequence [C_0, C_1, ...] where C_i is the number of irreducible characters of G with degree p^i for the finite p-group G*ì*Âjûÿûÿûÿûÿ?È>?´0Ã
__sig__31l2¼3ÄÈ>?ûÿÿÿŒ6Å*/usr/local/Magma2.8/package/Graph/graphs.mˆ­	`®	Æ*/usr/local/Magma2.8/package/Graph/graphs.mÐ­	`®	ÇThe n-th odd graph O_n|=¯®žssûÿûÿûÿ€><5t>¬Ÿ

__sig__112$ <5|>ûÿÿÿÐ¡NilpotentSectionØð¢ÿÿÿÿÿÿÿÿx>£*/usr/local/Magma2.8/package/AlgChtr/chtr.m¯	´	¤*/usr/local/Magma2.8/package/AlgChtr/chtr.mØ¯	´	¥:ÍRemove occurrences of the irreducibles in Irr from the characters in Chs and look for new irreducibles in the reduced set. Return a sequence of new irreducibles found and the sequence of reduced charactersT¦ÿJÿJûÿûÿûÿ >>”>§
__sig__0 !,!¨>œ>ûÿÿÿH#©RemoveIrreducibles $¸$ªÿÿÿÿÿÿÿÿ˜>«4³Make the norms of the characters in Chs smaller by computing the differences of appropriate pairs. Return a sequence of new irreducibles found and a sequence of reduced charactersÜ3ô3¬ÿJÿJûÿûÿûÿ¸>>¬>Ü9­
__sig__1:ä:ü:®>´>ûÿÿÿ\<üÿÿÿÿÿÿÿÿà?þaûÿûÿûÿûÿ@?ô?¨ÿ
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GewirtzGraphdÿÿÿÿÿÿÿÿø?The Cayley graph of the group GÀØðüÿìÿûÿûÿûÿûÿ@?@ 
	__sig__10ð?@ûÿÿÿ`
CayleyGraphh€ÿÿÿÿÿÿÿÿ@	-The Schreier graph for group G and subgroup H
üÿüÿìÿûÿûÿûÿûÿ0@?$@Ä#
	__sig__11ü$%?,@ûÿÿÿ|'
SchreierGraph)ÿÿÿÿÿÿÿÿ(@(/usr/local/Magma2.8/package/Sys/mutate.m0¸	ˆ¹	(/usr/local/Magma2.8/package/Sys/mutate.m6P¸	ˆ¹	Replace x with (x join y)è8 9ÿÿûÿûÿûÿûÿûÿP@@@D@`<
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__sig__1>À>à>¯ReduceCharacters>ˆ>¨>Ìÿÿÿÿÿÿÿÿ ?Í The triangular graph T_n (n > 1)t
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Îaûÿûÿûÿûÿ@??4?Ð?<?ûÿÿÿÀÑ
TriangularGraphDÒÿÿÿÿÿÿÿÿ8?ÓThe n-th square lattice graphLÔaûÿûÿûÿûÿX??L?LÕ
__sig__2\|Ö?T?ûÿÿÿÜ×SquareLatticeGraphØÿÿÿÿÿÿÿÿP?ÙBThe Paley graph of n; n must be a prime power congruent to 1 mod 4D!˜!Úaûÿûÿûÿûÿp??d?@%Û
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PaleyGraph,,,Þÿÿÿÿÿÿÿÿh?ßGThe Paley tournament of n; n must be a prime power congruent to 3 mod 4\6t6Œ6àbûÿûÿûÿûÿˆ??|?ü:á
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PaleyTournamentÜ>w
__sig__5\„xpAÜAûÿÿÿ˜y+/usr/local/Magma2.8/package/RngInvar/misc.mÐ¿	xÅ	z
+/usr/local/Magma2.8/package/RngInvar/misc.mÀ	xÅ	cldariants{,–The i-th Steenrod operation P^i(f) of f (f must be a multivariate polynomial with coefficients in a finite field and i must be a non-negative integer)L|KKûÿûÿûÿûÿøAèAìAL}
__sig__0\|~èAôAûÿÿÿÜÇ+/usr/local/Magma2.8/package/FldFun/places.mðÁ	È	È"iSequence of tuples of ramification indices and residue degrees of the places of F lying over the place P ìÉàAûÿûÿûÿûÿ<C,CC CÊ
__sig__04ËC(Cûÿÿÿ`ÌDecompositionTypeh€Íÿÿÿÿÿÿÿÿ$CÎ&{Sequence of tuples of ramification indices and residue degrees of the prime ideals of O lying over the prime polynomial p. ¨ÀÏâûÿûÿûÿûÿLCDCC8C@Ð
__sig__1 l ÑC@Cûÿÿÿ"Ò$tSequence of tuples of ramification indices and residue degrees of the prime ideals of the infinite maximal order O. (H(ÓâûÿûÿûÿûÿTCCHC.Ô
__sig__2080p0ÕCPCûÿÿÿÀ3Ö)/usr/local/Magma2.8/package/FldFun/zeta.m˜Æ	È	×)/usr/local/Magma2.8/package/FldFun/zeta.màÆ	È	Ø1The Zeta function of the global function field F Ø<ð<ÙÖÿSûÿûÿûÿûÿtClC\C`C8@SctionBSteenrodOperationx€ÿÿÿÿÿÿÿÿðA*Return whether invariant ring R is modular!0!‚ÙÿûÿûÿûÿûÿBèAB%ƒ
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	IsModularè+,†ÿÿÿÿÿÿÿÿB‡&A free resolution of (the module of) RÀ3ˆÙÿûÿûÿûÿûÿ(BèAB 9‰
__sig__2:¸:Ð:ŠèA$Bûÿÿÿ0<—
__sig__0ion =‹.A minimal free resolution of (the module of) R¨?__sig__3AHAhAˆAÀAbÿÿÿÿÿÿÿÿxAcJDisplay the Burnside matrix corresponding to the lattice of subgroups of G„œdæûÿûÿûÿûÿûÿ A˜ApAŒAde
__sig__1 ÈfpA”AûÿÿÿôgJDisplay the Burnside matrix corresponding to the lattice of subgroups of Gpˆhûÿûÿûÿûÿûÿ¨ApAœAÀi
__sig__2ÈàjpA¤Aûÿÿÿ kBThe Burnside matrix corresponding to the lattice of subgroups of G´læûÿûÿûÿûÿÈA¸ApA¬AL#npA´Aûÿÿÿ¼$o
BurnsideMatrixÔ&pÿÿÿÿÿÿÿÿ°AqBThe Burnside matrix corresponding to the lattice of subgroups of G/p/rûÿûÿûÿûÿØAÐApAÄA`6s
__sig__478p8tpAÌAûÿÿÿP:u?The Burnside matrix corresponding to the lattice of subgroups LÀ<Ø<ð<v´ûÿûÿûÿûÿàApAÔA8@Rÿÿÿÿÿÿÿÿ8AS'/usr/local/Magma2.8/package/Sys/names.m`Ò	T'/usr/local/Magma2.8/package/Sys/names.m`Ò	m
__sig__3trix	`Ò	@@L@ûÿÿÿtReplace x with (x meet y)Üüÿÿûÿûÿûÿûÿûÿ`@@@T@`@@\@ûÿÿÿReplace x with (x diff y)\tÿÿûÿûÿûÿûÿûÿp@@@d@
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__sig__6=ô=>,@@¬@ûÿÿÿ¤?¼?Ô?FÿÿÿÿÿÿÿÿAGReplace x with (x or y)°à	Hÿÿûÿûÿûÿûÿûÿ(A@@AdI
	__sig__13<TJ@@$AûÿÿÿLÿÿÿÿÿÿÿÿ AMReplace x with (x xor y) Nÿÿûÿûÿûÿûÿûÿ@A@@4AðO
	__sig__14(@P@@<Aûÿÿÿ˜UAssign Q to the names of S Vûÿûÿûÿûÿûÿûÿûÿ`APATAÜW
__sig__0 Ô !XPA\Aûÿÿÿœ"YSysAssignNamesNumT$l$ZÿÿÿÿÿÿÿÿXA[0/usr/local/Magma2.8/package/SubGrpLat/burnside.m+Ü	 Þ	\0/usr/local/Magma2.8/package/SubGrpLat/burnside.m0XÜ	 Þ	]GDisplay the Burnside matrix corresponding to the lattice of subgroups L6è607^´ûÿûÿûÿûÿûÿA€ApAtA0;_
__sig__0<H<`<`pA|AûÿÿÿÐ=aDisplayBurnsideMatrixx?¨pBœBûÿÿÿÔ©InternalFixSecondarySubmoduleHªÿÿÿÿÿÿÿÿ˜B«0/usr/local/Magma2.8/package/RngInvar/relations.m¨Þ	ä	¬0/usr/local/Magma2.8/package/RngInvar/relations.møÞ	ä	°,Algebraic relations for the invariant ring R\±ÙÿûÿûÿûÿûÿÄCÌB°BÀBü²
__sig__0$L³°BÈBûÿÿÿ¼´9The ideal of algebraic relations for the invariant ring RµÙÿLûÿûÿûÿûÿÜB°BÐBØ¶
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__sig__1@@4@üROOTSieldŒÙÿûÿûÿûÿûÿ8BèA,BŽèA4BûÿÿÿThe homological dimension of RÙÿûÿûÿûÿûÿHBèA<BáëJ;‘
__sig__4’èADBûÿÿÿ“+/usr/local/Magma2.8/package/RngInvar/prim.m å	pñ	”+/usr/local/Magma2.8/package/RngInvar/prim.mèå	pñ	Ÿ__sig__0variantsHard'—[;•Internal function–Ùÿûÿûÿûÿûÿ`BPBTB$˜PB\BûÿÿÿÚ
__sig__0`ˆÛ\ChCûÿÿÿœÜ]The Zeta function of the constant field extension of degree m of the global function field F ÝÖÿSûÿûÿûÿûÿÜ||C\CpCÞ
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__sig__0@å„CCûÿÿÿ´ æ2The denominator of a with respect to the order O. ¬#Ä#çâáþÿûÿûÿûÿûÿ¤C„C˜CH(è
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__sig__9quenceàSHReturns the three maps in the center of the exact sequence 0 -> k -> F -> Div -> Cl -> 0 where k is the global unit group of the function field, F is the multiplicative group of the function field, Div is the divisor group and Cl is the divisor class group

TÖÿûÿûÿÀFXE8DLE@
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WeilRestrictionÿÿÿÿÿÿÿÿD	,’A hyperelliptic function field in the Weil restriction over GF(q) of the elliptic function field E: y^2 + xy + x^3 + ax^2 + b defined over GF(q^n)d
ààXûÿûÿûÿ0DD$D
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ClassGroupors2«The number of effective divisors of degree less equal n who consist of places of degree less equal m only. P[i] contains the number of places of degree 1 <= i <= min(n, m))þÿûÿûÿûÿûÿHD8D<D
__sig__0D*8DDDûÿÿÿNumberOfSmoothDivisors/×Q3ÿÿÿÿÿÿÿÿ@D__sig__0ø'ðCTDÄvFToIdGIdToCl
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__sig__1d08D¼Dûÿÿÿ1ClassGroupAbelianInvariantsŒ>2ÿÿÿÿÿÿÿÿ¸D3$pThe divisor class group of F as an Abelian group and the map from the divisor group onto the divisor class groupd2;68DÔDûÿÿÿaIsSUnitMapœ*/usr/local/Magma2.8/package/RngInvar/sec.m˜û	àþ	:Secondary invariants for invariant ring R using subgroup Hu2žüÿÙÿûÿûÿûÿûÿˆB€BpBtBd pB|Bûÿÿÿ¡)Secondary invariants for invariant ring R÷`É:¢ÙÿûÿûÿûÿûÿBpB„Bd£
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ó„CÈCûÿÿÿ˜ô&|The exact constant field of F, i.e. the algebraic closure in F of the constant field of F, together with the inclusion map. õÖÿýÿûÿûÿûÿÜC„CÐC|ö
__sig__4l„÷„CØCûÿÿÿìøExactConstantField\|ùÿÿÿÿÿÿÿÿÔCú*/usr/local/Magma2.8/package/FldFun/roots.mH	à	û*/usr/local/Magma2.8/package/FldFun/roots.m	à	ý@Compute the roots of f which lie in the Riemann-Roch space of D #`##þ?ûÿûÿûÿûÿ`DìCôC(ÿ
__sig__0)\*t*ìCüCûÿÿÿ¬,·
__sig__0.8/package/,/usr/local/Magma2.8/package/FldFun/weilres.m	à	,/usr/local/Magma2.8/package/FldFun/weilres.m	à	,’A hyperelliptic function field in the Weil restriction over GF(q) of the elliptic function field E: y^2 + xy + x^3 + ax^2 + b defined over GF(q^n)„>¤>ààXûÿûÿûÿ(DDDDCžâûÿûÿûÿûÿÀ}€F8DtFHŸ
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	__sig__22äü¤8DŒFûÿÿÿ|¥PrincipalIdealMapÔì¦ÿÿÿÿÿÿÿÿˆF§LTrue and a generator if the fractional ideal I is principal, false otherwise<¨äáûÿûÿûÿÜ—¨F8DœF˜"©
	__sig__23 $8$ª8D¤Fûÿÿÿ`&«,’The ideal class group of the finite maximal order O as an Abelian group and the map from the group of fractional ideals onto the ideal class groupÄ/ü/¬â“ûÿûÿûÿ„˜¸F8D¬FŒ6­
	__sig__24„8œ8®8D´Fûÿÿÿœ:¹
	__sig__27nvariants°âûÿûÿÈF8D¼FÜ>±
	__sig__25ì?@²8DÄFûÿÿÿ¤Bµ
	__sig__26üDEpÿAûÿûÿûÿûÿÈE8D¼E¼
r8DÄEûÿÿÿhsSPrincipalDivisorMaptÿÿÿÿÿÿÿÿÀEuJTrue and a generator if D is principal modulo places in S, false otherwiseD\vÿA?áûÿûÿûÿàE8DÔEüx8DÜEûÿÿÿ|zÿÿÿÿÿÿÿÿØE{$uThe S-class group as an Abelian group and the map from the group of divisors (mod places in S) onto the S-class group¼|ÿA“ûÿûÿûÿøE8DìEh"}
	__sig__15Ø#$~8DôEûÿÿÿp%
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	__sig__17quence-DõReturns the three maps in the center of the exact sequence 0 -> U -> F -> Div -> Cl -> 0 where U is the S-unit group, F is the multiplicative group of the function field, Div is the group of divisors (mod places in S) and Cl is the S-class groupÌ;‚ÿAûÿûÿF8DFÜ>ƒ
	__sig__16ì?@„8DFûÿÿÿ¤B…SClassGroupExactSequenceE\E„EZÿA“ûÿûÿûÿpE8DdE0]'True if a is an S-unit, false otherwisePT^ÿAáûÿûÿûÿûÿ€E8DtEˆ_
	__sig__10œ`8D|Eûÿÿÿ´bÿÿÿÿÿÿÿÿxEáCentressscQTrue and the preimage of a in the S-unit group if a is an S-unit, false otherwise´dÿAá”ûÿûÿûÿ˜E8DŒEèe
	__sig__11üf8D”Eûÿÿÿ°	gIsSUnitWithPreimageÈ	hÿÿÿÿÿÿÿÿEiThe S-Regulatorü	
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SRegulatorà nÿÿÿÿÿÿÿÿ¨Ew
	__sig__14apØ$o fThe map from the multiplicative group of the function field to the group of divisors (mod places in S)XDq
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__sig__2Invariants-6The Abelian invariants of the divisor class group of FBÀ2The (complete) partial fraction decomposition of fÁSûÿûÿûÿûÿCäBCÃäBCûÿÿÿÄPartialFractionDecompositionÅÿÿÿÿÿÿÿÿCö˜GÔGûÿÿÿ÷The coeffient ring of fB€Y8øýÿûÿûÿûÿûÿ¨pèG˜GÜGB€Y8ú˜GäGûÿÿÿû'/usr/local/Magma2.8/package/RngPol/sd.mÀ(	ü'/usr/local/Magma2.8/package/RngPol/sd.mÀ(	
__sig__0ynomialý0The n-th SwinnertonDyer polynomial of degree 2^nÌþûÿûÿûÿûÿHðGôGÿ
__sig__0ðGüGûÿÿÿ`,5ûÿûÿûÿûÿØH¸HpH¬HL/SmallGroupIsSoluble(0ÿÿÿÿÿÿÿÿ°H7
__sig__5ble$1DReturn true if SmallGroup(o, n) is soluble (does not load the group)42ûÿûÿûÿûÿÐHpHÄH°3
__sig__4 À4pHÌHûÿÿÿP5DReturn true if SmallGroup(o, n) is soluble (does not load the group)6ûÿûÿûÿûÿàHpHÔHX8pHÜHûÿÿÿE
__sig__8uble0!9FReturn true if SmallGroup(o, n) is insoluble (does not load the group)¼$:ûÿûÿûÿûÿIðHpHäHx*;
__sig__6,,0,<pHìHûÿÿÿÀ.=SmallGroupIsInsolubleH1>ÿÿÿÿÿÿÿÿèH?IReturn true if SmallGroup(D, o, n) is insoluble (does not load the group)¸:Ð:@5ûÿûÿûÿûÿIpHüHÐ=A
__sig__7>0?H?BpHIûÿÿÿ0ACMThe size and number of the group isomorphic to G in the small groups databaseèEFpHIûÿÿÿ(HSwinnertonDyerPolynomialÄ(ÿÿÿÿÿÿÿÿøG2/usr/local/Magma2.8/package/Grp/is_conj_subgroup.mø#	À(	2/usr/local/Magma2.8/package/Grp/is_conj_subgroup.mH$	À(	MWhether a conjugate of N is a subgroup of M, and if so, a conjugating element2üÿüÿüÿöÿûÿûÿûÿ HHHdHHûÿÿÿ	IsConjugateSubgroup
ÿÿÿÿÿÿÿÿH+The set of conjugates of H by elements of G
š*1üÿüÿûÿûÿûÿûÿ8HH,Hd
__sig__1 t6H4HûÿÿÿÿÕÈ8+The set of conjugates of H by elements of GäAüÿüÿûÿûÿûÿûÿHHH<H
__sig__2CHDHûÿÿÿ€C,/usr/local/Magma2.8/package/Grp/commutator.m(	À(	,/usr/local/Magma2.8/package/Grp/commutator.m(	À(	çSolvableRadicaløüêDirectSumDecompositionÀèCartanSubalgebra,ìIsRestrictedLieAlgebraÀÁÂë
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KillingMatrixÈâº»¼½¾¿ÀÁÂÃÄÅÆÇÈñ+/usr/local/Magma2.8/package/AlgLie/cartan.mÀ*	À.	ú
__sig__1gebraÀòETrue if a Lie algebra of characteristic p > 0 is a restricted algebraðóÙûÿûÿûÿûÿÔKÄKÈKÀ	ô
__sig__0	Ô	õÄKÐKûÿÿÿì	ø*The Cartan subalgebra of the Lie algebra L
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ýÿÿÿÿÿÿÿÿäKþ+/usr/local/Magma2.8/package/AlgLie/center.mp-	À.	ÿ+/usr/local/Magma2.8/package/AlgLie/center.m¸-	À.	The centre of the Lie algebra L€5´BÙÙûÿûÿûÿûÿLüKLŒK¯LReturns the maps in the center of the exact sequence 0 -> U -> F -> Id -> Cl -> 0 where U is the unit group of O, F is the multiplicative group of the field of fractions of O, Id is the group of fractional ideals of O and Cl is the class group of O for a finite maximal order O°³MThe Abelian invariants of the ideal class group of the finite maximal order OŒ´âûÿûÿûÿûÿØF8DÌF<¶8DÔFûÿÿÿ¼·@The order of the ideal class group of the finite maximal order Od|¸âûÿûÿûÿûÿèF8DÜF¼º8DäFûÿÿÿh »=A sequence of fundamental units of the finite maximal order OÀ#¼âûÿûÿûÿûÿøF8DìFD(½
	__sig__28œ*¼*¾8DôFûÿÿÿì,¿FundamentalUnits/Ä/ü/ÀÿÿÿÿÿÿÿÿðFÁ=A sequence of independent units of the finite maximal order O9ÂâûÿûÿûÿûÿG8DG\<Ã
	__sig__29T=|=Ä8DGûÿÿÿD?ÅRemove all attributes from RAdAÆýÿûÿûÿûÿûÿûÿ G8DGF†ÿÿÿÿÿÿÿÿF	__sig__18Invariantsü‡+The Abelian invariants of the S-class group€˜ˆÿAûÿûÿûÿûÿ(F8DF`Š8D$Fûÿÿÿ‹SClassGroupAbelianInvariants\Œÿÿÿÿÿÿÿÿ F2The order of the torsion part of the S-class group$LŽÿAûÿûÿûÿûÿ@F8D4FD8D<FûÿÿÿŒ‘
SClassNumberè’ÿÿÿÿÿÿÿÿ8F“ eThe unit group of a finite maximal order O as an Abelian group and the map from the unit group into O¸$”â“ûÿûÿûÿÌðXF8DLFt*•
	__sig__19,,,–8DTFûÿÿÿ¼.—QTrue and the preimage of a in the unit group of O if a is a unit, false otherwiseD6\6˜ã”ûÿûÿûÿhF8D\FÌ:™
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)/usr/local/Magma2.8/package/RngPol/misc.mà<	(A	çLCM of elements of Q èÿþÿþÿûÿûÿûÿûÿ¨G˜GœGÜé
__sig__0 Ô !ê˜G¤Gûÿÿÿœ"ë*Generates a polynomial with coefficients L,%D%ìÿþÿûÿûÿûÿûÿÀG¸G˜G¬Gð*í
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__sig__2:¸:Ð:ò˜GÄGûÿÿÿ0<ó8Generates a polynomial over R by coercing the coeffients>À>à>ôýÿûÿûÿûÿûÿ\ØG˜GÌG¼Cõ
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	__sig__31œ¼Î8D4GûÿÿÿL5The commutator group [H, K] = <(h,k): h in H, k in K>tüÿüÿüÿûÿûÿûÿûÿ`HPHTHl
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__sig__0Ìì pH|HûÿÿÿÜ!DReturns the group number n of order o from the small groups database"L#"üÿûÿûÿûÿûÿ˜JHpH„H|'#
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__sig__2;X;€;(pHœHûÿÿÿÀ<)SmallGroupIsSolvable>p>*ÿÿÿÿÿÿÿÿ˜H+GReturn true if SmallGroup(D, o, n) is soluble (does not load the group)ˆE E¸E.pH´Hûÿÿÿ`GÏClassGroupChecks(Ðÿÿÿÿÿÿÿÿ0GÑ0/usr/local/Magma2.8/package/FldFun/weierstrass.mK	èN	Ò0/usr/local/Magma2.8/package/FldFun/weierstrass.m!XK	èN	Ó<Whether the degree one place P is a Weierstrass place of D. $l$ÔA?
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ûÿûÿûÿûÿˆGpGHGdGð<Û
__sig__1>H>p>ÜHGlGûÿÿÿØ?Ý bThe sequence of gap numbers of the function field of P at P where P must be a place of degree one ÐEèEDüÿûÿûÿûÿûÿIpHIHG-The number of groups stored in the database DÔH5ûÿûÿûÿûÿ8I(IpHIÀI
__sig__9,DJpH$Iûÿÿÿ|ûÿûÿûÿûÿûÿøIpHìIL InternalSmallGroupProcessRestart¨€ÿÿÿÿÿÿÿÿðIxÃReturns a small group process. This will iterate through all groups with order in Orders. To extract the current group from a process, use ExtractGroup(). To move to the next group in a process, use NextGroup(). To find out which group the process currently points to, use ExtractLabel(). The user may limit the process to soluble or insoluble groups by setting Search. Search may take the values "All", "Soluble" or "Insoluble" (or variants thereof).ø‚ÿûÿûÿûÿûÿ JJpHJ´ „pHJûÿÿÿl"…SmallGroupProcess$$<$†ÿÿÿÿÿÿÿÿJ‡*‹Returns a small group process as described above. This will iterate through all groups (g) with order in Orders which satisfy Predicate(g).80p0ˆãÿÿûÿûÿûÿûÿ0J(JpHJ07‰
	__sig__21è8 9ŠpH$JûÿÿÿÐ:‹"hReturns a small group process as described above. This will iterate through all groups (g) with order o.>À>à>Œûÿûÿûÿûÿ@J8JpH,J¼C
	__sig__22`EˆEŽpH4Jûÿÿÿ(G‘
	__sig__230IxIKNumberOfSmallGroups Lÿÿÿÿÿÿÿÿ IM;The number of groups stored in the database of small groups`€NûÿûÿûÿûÿHI@IpH4IXO
	__sig__10d„PpH<IûÿÿÿQ8The number of groups of order n stored in the database D"4"l"R5ûÿûÿûÿûÿXIPIpHDIÔ&S
	__sig__11`(x(TpHLIûÿÿÿ¸+UFThe number of groups of order n stored in the database of small groups¸0Vûÿûÿûÿûÿ`IpHTI7W
	__sig__12h9 9XpH\Iûÿÿÿ;e
	__sig__15Limit0<Y=The order of the largest group in the small groups database Dà>Z5ûÿûÿûÿûÿ€IpIpHdI¼C[
	__sig__13`EˆE\pHlIûÿÿÿ(G¥BíReturns the first group (g) of order o which satisfies Predicate. The user may limit the the search to soluble or insoluble groups by setting the parameter Search. Search may take the values "All", "Soluble" or "Insoluble" (or variants).T¦ãÿ5üÿûÿûÿûÿûÿ¨J JpH”J”©BíReturns the first group (g) of order o which satisfies Predicate. The user may limit the the search to soluble or insoluble groups by setting the parameter Search. Search may take the values "All", "Soluble" or "Insoluble" (or variants).àªãÿüÿûÿûÿûÿûÿ¸J°JpH¤Jì«
	__sig__28´¬pH¬Jûÿÿÿ!­*ˆReturns the first group (g) with order in Orders which satisfies the predicate and the search condition specified by Search (see above).'ì'(®ãÿüÿûÿûÿûÿûÿÈJÀJpH´J-¯
	__sig__29È/0°pH¼Jûÿÿÿ¨1±*ˆReturns the first group (g) with order in Orders which satisfies the predicate and the search condition specified by Search (see above).;;0;²ãÿ5üÿûÿûÿûÿûÿØJÐJpHÄJp>³
	__sig__30?¨?´pHÌJûÿÿÿÀAµ"lReturns the first group (g) of order o which satisfies the search condition specified by Search (see above).H(H·
	__sig__31ÐIèI]SmallGroupDatabaseLimit^ÿÿÿÿÿÿÿÿhI_;The order of the largest group in the small groups databaseèC`ûÿûÿûÿûÿˆIpH|Ia
	__sig__14bpH„IûÿÿÿLHc?Return the small groups database, opened for an extended searchÈð8d5ûÿûÿûÿûÿ˜IpHŒIÀ
fpH”IûÿÿÿlgSmallGroupDatabase|”hÿÿÿÿÿÿÿÿI}
	__sig__19roupÈi?Return the small groups database, opened for an extended searchð j5ûÿûÿûÿûÿ°IpH¤Iðk
	__sig__16(@lpH¬Iûÿÿÿ˜mOpenSmallGroupDatabaseðnÿÿÿÿÿÿÿÿ¨IoClose the smallgroup databasep5ûÿûÿûÿûÿûÿÈIpH¼IÄ#q
	__sig__17ü$%rpHÄIûÿÿÿ|'sCloseSmallGroupDatabasex*tÿÿÿÿÿÿÿÿÀIu7Moves the small group process tuple p to its next groupx11¨1vûÿûÿûÿûÿûÿàIpHÔI 9w
	__sig__18x: :xpHÜIûÿÿÿ<yInternalNextSmallGroup =zÿÿÿÿÿÿÿÿØIƒ	__sig__20ProcessRestartA{:Returns the small group process tuple p to its first group E¸E~pHôIûÿÿÿ`Gà+/usr/local/Magma2.8/package/AlgLie/cartan.møj	0z	ã
DerivedSeriesB€Y8ˆDual¼äLowerCentralSeriesdåUpperCentralSeriesDé
LeviSubalgebra¾æ

NilRadicalP—DegreesOfGenerators˜ÿÿÿÿÿÿÿÿPR×__sig__5ionutionp@™V:Given projective resolutions P and Q for simple module S and T over a basic algebra A and the cohomology generators for T associated to the resolution Q, the function returns the chain maps in compact form of the minimal generators for the cohomology Ext*(S,T) as a right module over the cohomology ring Ext*(T,T).ìšûÿûÿûÿûÿpR0RdR¸œ0RlRûÿÿÿÐCohomologyRightModuleGeneratorsìžÿÿÿÿÿÿÿÿhRŸV9Given projective resolutions P and Q for simple module S and T over a basic algebra A and the cohomology generators for T associated to the resolution Q, the function returns the chain maps in compact form of the minimal generators for the cohomology Ext*(S,T) as a left module over the cohomology ring Ext*(S,S).T
 ûÿûÿûÿûÿˆR0R|Rt!¡
__sig__3$Ì$¢0R„Rûÿÿÿä$£CohomologyLeftModuleGeneratorsp5¤ÿÿÿÿÿÿÿÿ€R¥IReturns the degrees of the chain maps of the generators of the cohomology˜K¦ûÿûÿûÿûÿ R0R”RRÿÿÿÿÿÿÿÿdL*Test whether the Lie algebra L is solvable¤
¼
Ùûÿûÿûÿûÿ„LüKxLÀ 
__sig__7¨À!üK€Lûÿÿÿt"+Test whether the Lie algebra L is nilpotentD\#ÙûÿûÿûÿûÿÔL”LüKˆLü$
__sig__8$L%üKLûÿÿÿ¼&+/usr/local/Magma2.8/package/AlgLie/nilrad.mu	ˆy	'+/usr/local/Magma2.8/package/AlgLie/nilrad.mHu	ˆy	(#The nilradical of the Lie algebra L@è)ÙÙûÿûÿûÿûÿ¬LœL L$*
__sig__0%(%@%+œL¨Lûÿÿÿ(,

NilRadical\*t*-ÿÿÿÿÿÿÿÿ¤L.(The soluble radical of the Lie algebra L11D1/ÙÙûÿûÿûÿûÿÄLœL¸Ll80
__sig__19ô9:1œLÀLûÿÿÿ|;2,/usr/local/Magma2.8/package/AlgLie/nonnilp.mx	ˆy	3,/usr/local/Magma2.8/package/AlgLie/nonnilp.mx	ˆy	47True if the element x of the Lie algebra L is nilpotentDLDÄD5ðÙûÿûÿûÿûÿÜLÌLÐLôH6
__sig__0J\JtJöIsRestrictedLieAlgebrad÷ÿÿÿÿÿÿÿÿÌKûÄKèKûÿÿÿ(Returns a small group process as described above. This will iterate through all groups (g) of order o which satisfy Predicate(g).@hãÿûÿûÿûÿûÿHJpH<JT’pHDJûÿÿÿ§	__sig__27llGroupLabely“HReturns true if the small group process tuple has passed its last group.Èà”ûÿûÿûÿûÿXJpHLJ•
	__sig__24(–pHTJûÿÿÿ€— InternalSmallGroupProcessIsEmpty8L˜ÿÿÿÿÿÿÿÿPJ™<Returns the current group of the small group process tuple p"œ"šüÿûÿûÿûÿûÿpJpHdJ'›
	__sig__25(¨(œpHlJûÿÿÿ,InternalExtractSmallGrouph.À.žÿÿÿÿÿÿÿÿhJŸ$rReturns the label (s,n) of the small group process tuple p. This is the order and number of the current group of px: : ûÿûÿûÿˆJpH|J€=¡
	__sig__26À>à>¢pH„Jûÿÿÿ8@
__sig__0\„üKLûÿÿÿ˜/The centralizer of ideal K in the Lie algebra LdŒ¤ÙÙÙûÿûÿûÿûÿLüKL
__sig__1üüKLûÿÿÿ`.The normalizer of ideal K in the Lie algebra L	ÙÙÙûÿûÿûÿûÿ,LüK Lì

__sig__2$<üK(Lûÿÿÿ”'The derived series of the Lie algebra LTtŒÙûÿûÿûÿûÿ<LüK0Lè
__sig__3!D!˜!üK8Lûÿÿÿ#-The lower central series of the Lie algebra L`&ÙûÿûÿûÿûÿLLüK@Lü+
__sig__4-ä-ü-üKHLûÿÿÿ´0-The upper central series of the Lie algebra L,6Ùûÿûÿûÿûÿ\LüKPLœ:
__sig__5;¤;Ì;üKXLûÿÿÿì<FThe adjoint matrix corresponding to the element x of the Lie algebra L@ðÙûÿûÿûÿûÿ(ulLüK`L„E
__sig__6FlF”FüKhLûÿÿÿtI
AdjointMatrixK£InternalExtractSmallGroupLabelDE¤ÿÿÿÿÿÿÿÿ€J¨pHœJûÿÿÿäI<NonNilpotentElement°Ø=ÿÿÿÿÿÿÿÿäL>./usr/local/Magma2.8/package/AlgLie/semsimalg.m	xŽ	?./usr/local/Magma2.8/package/AlgLie/semsimalg.m	xŽ	@Create a classical Lie algebra 0HAõÿÙûÿûÿûÿûÿMüLMðB
__sig__0ÈàCüLMûÿÿÿ@DSimpleLieAlgebra¨ÀEÿÿÿÿÿÿÿÿMF)/usr/local/Magma2.8/package/AlgLie/levi.mŠ	xŽ	G)/usr/local/Magma2.8/package/AlgLie/levi.mPŠ	xŽ	H&A Levi subalgebra of the Lie algebra L0!IÙÙûÿûÿûÿ,MM M%J
__sig__0&ì&'KM(Mûÿÿÿ)Mÿÿÿÿÿÿÿÿ$MN+/usr/local/Magma2.8/package/AlgLie/dirdec.mØ‹	xŽ	O+/usr/local/Magma2.8/package/AlgLie/dirdec.m Œ	xŽ	P#The matrix of the Killing form of Lh9 9QÙwûÿûÿûÿûÿLM<M@M<S<MHMûÿÿÿ >T
KillingMatrixx?UÿÿÿÿÿÿÿÿDM`
__sig__0itionÈDVEA list of ideals of the Lie algebra L such that L is their direct sum¨HX
__sig__1IJJFPushoutL¶üÿûÿûÿûÿûÿèJàJpHÔJ¸pHÜJûÿÿÿ¹"lReturns the first group (g) of order o which satisfies the search condition specified by Search (see above).º5üÿûÿûÿûÿûÿðJpHäJ»
	__sig__32¼pHìJûÿÿÿ½`cReturns a list of all groups of order o. The user may limit the search to soluble or insoluble groups by setting the parameter Search. Search may take the value "All", "Soluble" or "Insoluble" (or variants thereof). Some orders will produce a very large list of groups -- in such cases a warning will be printed unless the user specifies Warning := false.¾5ÅûÿûÿûÿûÿKKpHôJ¿
	__sig__33ÀpHüJûÿÿÿÁ
SmallGroupsÂÿÿÿÿÿÿÿÿøJÃ`cReturns a list of all groups of order o. The user may limit the search to soluble or insoluble groups by setting the parameter Search. Search may take the value "All", "Soluble" or "Insoluble" (or variants thereof). Some orders will produce a very large list of groups -- in such cases a warning will be printed unless the user specifies Warning := false.ÄÅûÿûÿûÿûÿ KKpHKÅ
	__sig__34ÆpHKûÿÿÿÇ$As above, but with a list of orders.È5Åûÿûÿûÿûÿ0K(KpHKÉ
	__sig__35ÊpH$KûÿÿÿË$As above, but with a list of orders.ÌÅûÿûÿûÿûÿ@K8KpH,KÍ
	__sig__36ÎpH4KûÿÿÿÏ&}Returns a list of all groups (g) with order o which satisfy Predicate(g) eq true and the search condition specified by SearchÐãÿ5ÅûÿûÿûÿûÿPKHKpH<KÑ
	__sig__37ÒpHDKûÿÿÿÓ&}Returns a list of all groups (g) with order o which satisfy Predicate(g) eq true and the search condition specified by SearchÔãÿÅûÿûÿûÿûÿ`KXKpHLKÕ
	__sig__38ÖpHTKûÿÿÿ×$As above, but with a list of orders.Øãÿ5ÅûÿûÿûÿûÿpKhKpH\KÙ
	__sig__39ÚpHdKûÿÿÿÛ$As above, but with a list of orders.ÜãÿÅûÿûÿûÿûÿxKpHlKÝ
	__sig__40ÞpHtKûÿÿÿß+/usr/local/Magma2.8/package/AlgLie/attrib.mð›	h¢	7ÌLØLûÿÿÿ8,A non-nilpotent element of the Lie algebra Ld9ÙðûÿûÿûÿûÿìLÌLàL:
__sig__1dB€Y8;ÌLèLûÿÿÿtWCreate the complex given by the list L of maps and such that the last term has degree d¬ÜüuÅ8ûÿûÿûÿûÿÜMÌMÐM`v
__sig__08PwÌMØMûÿÿÿx<ÕThe splice of the complex C with the complex D along the map f from the last term of C to the first term of D. the degree of the last term of the splice is the same as the degree of the last term of the complex D.|yw888ûÿûÿûÿûÿìMÌMàM|z
__sig__1tŒ{ÌMèMûÿÿÿT„
__sig__3mplexØ|0The sequence of homology groups of the complex C"0"h"}8ûÿûÿûÿûÿüMÌMðMÐ&~
__sig__2(\(t(ÌMøMûÿÿÿ´+ˆ"hThe homology group in degree n of the complex C as an A-module, together with the associated epimorphism1l2¼3‰8ûÿûÿûÿûÿ,NÌM Nœ9Š
__sig__4:´:Ì:‹ÌM(Nûÿÿÿ,<Œ9The dimension of the n-th homology group of the complex C¼>Ü>8ûÿûÿûÿûÿ<NÌM0N¸CŽ
__sig__5E\E„EÌM8Nûÿÿÿ$GDimensionOfHomologyœI´I‘ÿÿÿÿÿÿÿÿ4NWÙûÿûÿûÿûÿlddM<MXM¬Y<M`Mûÿÿÿd£889ûÿûÿûÿ”NÌMˆNL¥ÌMNûÿÿÿ(¦RandomSubcomplex´Ô§ÿÿÿÿÿÿÿÿŒN¨?The quotient complex Q of C by D and the projection of C onto Q4Ld©888ûÿûÿûÿûÿ¼N¬NÌM Nà«ÌM¨Nûÿÿÿ@¬
QuotientComplex€­ÿÿÿÿÿÿÿÿ¤N®9The quotient complex of C by the sequence S of submodules¸à¯ÿ88ûÿûÿûÿûÿÌNÄNÌM¸Nl °
	__sig__11´!ì!±ÌMÀNûÿÿÿÄ#²P The quotient of C by the subcomplex generated by L. The list L may be either a list of sequences of element of the terms of C or a list of submodules of the terms of C. There must be one element of L for each term of C. The function returns both the quotient complex and the quotient map.4˜5ð5³Å889ûÿûÿûÿÔNÌMÈNP:´
	__sig__12X;€;µÌMÐNûÿÿÿÀ<¶The direct sum of C and Dˆ>¨>·888ûÿûÿûÿûÿ\däNÌMØNC¸
	__sig__130EHE¹ÌMàNûÿÿÿ˜F»99ûÿûÿûÿûÿôNÌMèN´L‰ÍZDirectSumDecompositiond[ÿÿÿÿÿÿÿÿ\M\//usr/local/Magma2.8/package/AlgLie/rootsystem.m0²	]//usr/local/Magma2.8/package/AlgLie/rootsystem.m0²	^+The root system of a semisimple Lie algebraè_Ùûÿ„MtMxM(atM€Mûÿÿÿ€b

RootSystemxcÿÿÿÿÿÿÿÿ|Md#The root datum of the Lie algebra LÔ !eÙØûÿûÿûÿûÿÄcœMtMM¼$f
__sig__1&Œ&Ô&gtM˜Mûÿÿÿ¨(h
	RootDatum+¸+iÿÿÿÿÿÿÿÿ”Mj)/usr/local/Magma2.8/package/AlgLie/type.m ¯	0²	k)/usr/local/Magma2.8/package/AlgLie/type.mè¯	0²	l$The type of a semisimple Lie algebra: :mÙûÿûÿûÿûÿ¼M¬M°M€=n
__sig__0>À>à>o¬M¸Mûÿÿÿ8@p
SemiSimpleType¨Bqÿÿÿÿÿÿÿÿ´Mr,/usr/local/Magma2.8/package/AlgBas/complex.m±	0²	sB,/usr/local/Magma2.8/package/AlgBas/complex.m±	0²	xy	ä»çLHasLeviSubalgebraœI´IR__sig__0KØKðKpL°LâDõGiven a sequence of complexes of modules over the same algebra, the function returns the sequence of complexes obtained by taking Zero extensions of the elements of S, if necessary until all of the elements of the sequence have the same degrees.”æ
EqualizeDegreesçÿÿÿÿÿÿÿÿŒOèDñGiven a complex C in degrees from a to b and a complex D in degrees c to d, the function returns the two complexes in degrees max(a,c+n) to min(b,d+n) and from max(a-n,c) to min(b-n,d) obtained from C and D by zero extensions when necessary.(Hé8888ûÿûÿûÿ´O¬OÌM Oê
	__sig__23ôëÌM¨Oûÿÿÿœ!ì6»Given complexes C and D over the same algebra, the function returns the complexes obtained by taking Zero extensions of C and D, if necessary so that both complexes have the same degrees.+¸+í8888ûÿûÿûÿ¼OÌM°O1î
	__sig__24à3ø3ïÌM¸Oûÿÿÿ07ð@æGiven a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adding the inclusion map from the homology to the term of highest degree in the complex to the left end of the complex.Ø?ñ88ûÿûÿûÿûÿÌOÌMÀOHEò
	__sig__250FHFóÌMÈOûÿÿÿøHôLeftExactExtensionJKõÿÿÿÿÿÿÿÿÄOº*ŠTakes a chain map in any degree and writes it as a chain map in degree zero in which both the domain and codomain have lowest degree zero.„œ¼
	__sig__14|”½ÌMðNûÿÿÿÄ¾ShiftToDegreeZero`x¿ÿÿÿÿÿÿÿÿìNÀ*‡The dual of the complex C as a complex over the opposite algebra of the algebra of C. The last term of the dual complex is in degree 0.€¨ÀÁ88ûÿûÿûÿûÿOOÌMOÀÂ
	__sig__15¸àÃÌMOûÿÿÿÄ*‡The dual of the complex C as a complex over the opposite algebra of the algebra of C. The last term of the dual complex is in degree n.<$T$l$Å88ûÿûÿûÿûÿèUOÌMO)Æ
	__sig__16è+,ÇÌMOûÿÿÿ0.È:ÈGiven a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adjoining the zero map from the zero module to the term of highest degree in the complex.:è:;É88ûÿûÿûÿûÿ<O,OÌM O >Ê
	__sig__17`?x?ËÌM(OûÿÿÿhAÌLeftZeroExtensionäC DÍÿÿÿÿÿÿÿÿ$OÎ]Returns the complex obtained from C by apending n terms of zero modules on the left end of C.MÐ
	__sig__18DN\Nš
__sig__7ogyx’FThe sequence of the dimensions of the homology groups of the complex Ch“8ûÿûÿûÿûÿTNÌMHNT”
__sig__6Œ´•ÌMPNûÿÿÿ4–DimensionsOfHomology`—ÿÿÿÿÿÿÿÿLN˜$oThe subcomplex of C generated by S and the inclusion of S in C. S is a sequence of submodules of the terms of Cð(™ÿ88ûÿûÿûÿûÿ|NlNÌM`N›ÌMhNûÿÿÿÀœ

SubcomplexDìÿÿÿÿÿÿÿÿdNžNThe subcomplex generated by L. The list L may be either a list of sequences of element of the terms of C or a list of submodules of the terms of C. There must be one element of L for each term of C. The function returns both the subcomplex and the inclusion of the subcomplex into C.`1x1ŸÅ889ûÿûÿûÿ„NÌMxN 8 
__sig__8:8:P:¡ÌM€NûÿÿÿÐ;¢X?Given a chain complex C in degrees a to a-t+1 and a sequence S = [s_1 .. s_t], the function creates the minimal chain comples whose term in degree a-i+1 is a subcomplex generated by s_i random elements of the term in degree a-i+1 of C. The function also returns the chain map that is the inclusion of subcomplex into C.¸IÐIèI¤
__sig__9KtL´Lª
	__sig__10NN€HomologyOfChainComplex,;ÿÿÿÿÿÿÿÿôM‚0The sequence of homology groups of the complex C?Œ?¤?ƒ8ûÿûÿûÿûÿ$NNÌMNE…ÌMNûÿÿÿDF†
HomologyHdH¤H‡ÿÿÿÿÿÿÿÿN
psnsion¿
__sig__1\„ÀØRüRûÿÿÿ˜ï~ÚCreates a basic algebra from a minimal set of relations. The input is a free algebra, the number of idempotents, a list of the right and left idempotents corresponding to the nonidempotent generators of the algebra and a list of relations. The function creates the extra relations that are natural to the idempotents and the multiplications of the idempotents on the nonidempotent generators. Thus the user needs only supply the relations among the nonidempotent generators.è÷The zero A-moduleh€øûÿûÿûÿûÿèS˜SÜS€ù
__sig__4xú˜SäSûÿÿÿXû

ZeroModuledÜüÿÿÿÿÿÿÿÿàSýThe zero map from M to N#Ü#$þwûÿûÿûÿûÿT˜SôSx(ÿ
__sig__5*Ø*ð*˜SüSûÿÿÿ-ÿÿÿÿÿÿÿÿøS+/usr/local/Magma2.8/package/AlgBas/mapchn.mXÎ	hÑ	(+/usr/local/Magma2.8/package/AlgBas/mapchn.m Î	hÑ	BCreate the chain map from C to D all of whose module maps are zerox<<889ûÿûÿûÿûÿ TTTØ?
__sig__0AˆAÀATTûÿÿÿÈDT4TûÿÿÿFIsProperChainMapHhH¨Hÿÿÿÿÿÿÿÿ0T0 True if the list of maps L from the terms of complex C to the terms of D is a chain complex of degree n, i. e. it has the right lenghth and the diagram commutesRS SÁSimpleCohomologyDimensions@hÂÿÿÿÿÿÿÿÿøRÃÂëGiven a module M over a basic algebra and a natural number n the function computes a projective resolution for M out to n steps. The function returns the resolution in compact form together with the augmentation map (P_0 -> M). The compact form of the resolution is a list of the the minimal pieces of information needed to reconstruct the boundary maps in the resolution. That is the boundary map (P_{i+1} -> P_i) is recorded as a tuple consisting of a matrix whose entries are the images of the generators for indecomposable projective modules making up P_{i+1} in the indecomposable projective modules making up P_i and two lists of integers givin the number of indecomposable projective modules of each isomorphism class in P_{i+1} and in P_i.((H(ÄûÿûÿûÿûÿSØRS.Å
__sig__2080p0ÆØRSûÿÿÿÀ3ÇCompactProjectiveResolutionè607ÈÿÿÿÿÿÿÿÿSÊûÿûÿûÿûÿ0SØR$Sð<Ë
__sig__3>H>p>ÌØR,SûÿÿÿØ?ÍCompactInjectiveResolutionhB¨BÎÿÿÿÿÿÿÿÿ(SÏFýThe complex giving the minimal projective resolution of M together with the augmentation homomorphism from the projective cover of M into M. Note that homomorphisms go from left to right so that the cokernel of the last homomorphism in the complex is M.äQZeroMapion¯CohomologyGeneratorToChainMap°ÿÿÿÿÿÿÿÿ°R²89ûÿûÿûÿûÿÐR0RÄR³
__sig__6dî 69´0RÌRûÿÿÿÏ88ûÿûÿûÿûÿDOÌM8OHÑÌM@Oûÿÿÿ$Ò:ÇGiven a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adjoining the zero map to the zero module from the term of lowest degree in the complex.ÀØðÓ88ûÿûÿûÿûÿdOTOÌMHO„Ô
	__sig__19„œÕÌMPOûÿÿÿ<ÖRightZeroExtensiont”×ÿÿÿÿÿÿÿÿLOØ^Returns the complex obtained from C by apending n terms of zero modules on the right end of C.h Ù88ûÿûÿûÿûÿlOÌM`O8$Ú
	__sig__20X%p%ÛÌMhOûÿÿÿD(ÜR)Given a complex C of modules over a basic algebra, return the complex of lenth two greater that is obtained by adjoining the zero map to the zero module from the term of lowest degree in the complex and also appending the zero map from the zero module to the term of highest degree in the complex.t:œ:Ý88ûÿûÿûÿûÿ|OÌMpO|=Þ
	__sig__21¼>Ü>ßÌMxOûÿÿÿ4@8ûÿûÿûÿûÿ4PÌM(PL
	__sig__30Ø
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ÌM0Pûÿÿÿœ/The list of the boundary maps in the complex C.T|”8ÅûÿûÿûÿûÿDPÌM8PÄ
	__sig__31¨ÀÌM@Pûÿÿÿà
__sig__0nceOTrue if the complex consists of a short exact sequence along 	 with other terms(@`8ûÿûÿûÿ@UTPÌMHPL
	__sig__32¨ÀÌMPPûÿÿÿl IsShortExactSequence""ÿÿÿÿÿÿÿÿLP-/usr/local/Magma2.8/package/AlgBas/algebras.m	ç	&-/usr/local/Magma2.8/package/AlgBas/algebras.m	ç	wûÿûÿûÿûÿtPdPhPH1dPpPûÿÿÿð5
MapToMatrixh77ÿÿÿÿÿÿÿÿlP(
__sig__2gBas€; The Jacobson radical of MØ<ð<!ûÿûÿûÿûÿŒPdP€P8@"
__sig__1BhB¨B#dPˆPûÿÿÿHE$JacobsonRadicalAlgBas˜F%ÿÿÿÿÿÿÿÿ„P'wûÿûÿûÿûÿÄP¤PdP˜PüNà
ZeroExtension¼CáÿÿÿÿÿÿÿÿtOãûÿûÿûÿûÿ¤O”OÌMˆOKä
	__sig__22lMŒMåÌMOûÿÿÿ\OöBìGiven a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adding the quotient map to the homology module from the term of lowest degree in the complex on the right end of the complex.P÷88ûÿûÿûÿûÿäOÌMØOø
	__sig__26tŒùÌMàOûÿÿÿ¬úRightExactExtensionäüûÿÿÿÿÿÿÿÿÜOüP&Given a complex C of modules over a basic algebra, return the complex of lenth two greater that is obtained by adding the inclusion from the homology to the term of highest degree in the complex and also appending the quotient map from the homology to the term of highest degree in the complex.#ý88ûÿûÿûÿûÿüOÌMðO(þ
	__sig__27\*t*ÿÌMøOûÿÿÿ¬,&>×Given an element x in a module over a basic algebra, creates the homomorphism from the nth projective module for the algebra to the module with the property that the idempotent e of the projective module maps to x*eÄÜô)dP Pûÿÿÿd*LiftHomomorphismì+ÿÿÿÿÿÿÿÿœP,NGiven a sequence S = [s_1,s_2, ... ], the function returns a projective module which is the direct sum of s_1 copies of the first projective of the algebra A, s_2 copies of the second, etc. It also returns the sequences of inclusions and projections onto the incecomposable projecitves.L-ûÿûÿ¼PdP°P!.
__sig__3"4"l"/dP¸Pûÿÿÿ<$0``Given an sequence X of elements in a module over a basic algebra and a sequence N of nonnegative integers, the function creates the homomorphism from the projective module that is a direct sum of N[i] copies of the ith projective module for the algebra to the module of X with the property that the idempotent e of the projective module maps to X[i]*e.9h9 91wûÿûÿûÿûÿÌPdPÀP<2
__sig__4=¨=Ð=3dPÈPûÿÿÿx?5wÜPdPÐPÈD6
__sig__5EÐEèE7dPØPûÿÿÿ(H8
ProjectiveCoverèI9ÿÿÿÿÿÿÿÿÔP<
__sig__6N´NüNƒÉÊ
ExactExtension0ÿÿÿÿÿÿÿÿôO=Returns the complex C with n terms removed from the left end. 988ûÿûÿûÿûÿPÌMP`<
	__sig__28X=€=ÌMPûÿÿÿH?>Returns the complex C with n terms removed from the right end.C88ûÿûÿûÿûÿ$PÌMP`G
	__sig__29 I¸I	ÌM Pûÿÿÿ´L
)The sequence of the terms of the complex.4O\O^"nTht algebra $A$ as a right module over itself. The module is the direct sum of the projectives modules of $A$.¼
_ûÿûÿûÿûÿ„QdPxQÀadP€QûÿÿÿðbRightRegularModuleœÄcÿÿÿÿÿÿÿÿ|Qr	__sig__16iveModules|dRThe sequence of the dimensions of the projective modules of the basic algebra $A$.¤¼eûÿûÿûÿûÿœQdPQ¼f
	__sig__14´ÜgdP˜QûÿÿÿŒhDimensionsOfProjectiveModulesh iÿÿÿÿÿÿÿÿ”QjQThe sequence of the dimensions of the injective modules of the basic algebra $A$.X'x'kûÿûÿûÿûÿ´QdP¨Qì,l
	__sig__15/l/mdP°Qûÿÿÿt1nDimensionsOfInjectiveModules6\6oÿÿÿÿÿÿÿÿ¬QpV7True if the module M is a semisimple module and false otherwise. If true, then the function also returns a list of the ranks of the primitive idempotents of the algebra. This is also a list of the multiplicities of the simple modules of the algebra as composition factors in a composition series for the module.äEüEFqûÿûÿûÿ\cÌQdPÀQKsdPÈQûÿÿÿÀMt4Returns the identity (n x n)-matrix over the ring R.PàPQ0QGÿÿÿÿÿÿÿÿQH&}Returns the pullback of the diagram [ M1 -- fc1 --> N <-- fc2 -- M2 ] as an AModule together with homomorphisms to M1 and M2.dIwwwwûÿûÿü‰,QdP QxJ
__sig__9|”KdP(QûÿÿÿÀL
Pullback˜°Mÿÿÿÿÿÿÿÿ$QN$rThe nth irreducible module of the algebra. The module is the quotient of the nth projective module by its radical.¨ÀOûÿûÿûÿûÿDQdP8Q@P
	__sig__10 l QdP@Qûÿÿÿ"RIrreducibleModuleÜ#$Sÿÿÿÿÿÿÿÿ<QT$rThe nth irreducible module of the algebra. The module is the quotient of the nth projective module by its radical..0.Uûÿûÿûÿûÿ\QdPPQð5V
	__sig__11h77WdPXQûÿÿÿà9X
SimpleModule:;YÿÿÿÿÿÿÿÿTQ`
	__sig__13ion >Z6½If $v$ is an element of a basic algebra given as a vector in the underlying space, then the function computes the matrix of the action by right multiplication of the element on the algebra.(G[ûÿûÿûÿûÿtQdPhQM\
	__sig__12DN\N]dPpQûÿÿÿ\P4^]The projective cover of M given as the projective module P, the surjective homomorphism of P onto M, the sequences of inclusion and projection homomorphism of P from and to its indecomposable direct summands and the isomorphism type of P in the form of a list of the number of copies of the projective modules of algebra of each type that make up P.:0¢Returns true if the module is projective and returns a list of how many projective modules of each type are direct summands of the projective cover of the module.;ûÿûÿûÿDœôPdPèP=dPðPûÿÿÿ>.žReturns true if the module is injective and returns a list of how many injective modules of each type are direct summands of the injective hull of the module.?ûÿûÿûÿUQdPøP@
__sig__7AdPQûÿÿÿB&~Returns the pushout of the diagram [ M1 <-- fc1 -- D --- fc2 --> M1 ] as an AModule together with homomorphisms from M1 and M2CwwwwûÿûÿQdPQD
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	RootNormsÓÔ·
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CoxeterGroupï½LeftStringLengthsÀÄ3The space containing the roots of the root datum RDÛÜÅØÿÿûÿûÿûÿûÿDh_l^_õÂStructureConstantsÐ	Ã

NatRepnMxsä	0*ÐÑÒÓÔÈÕÖ×Ø!ÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõöÆ
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X
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	RootSpacep!Éÿÿÿÿÿÿÿÿ_Ê5The space containing the coroots of the root datum RDl5ËØÿÿûÿûÿûÿûÿTh4_l^(_„DÌ
__sig__1D„KÍl^0_ûÿÿÿ KÎ
CorootSpace´KÏÿÿÿÿÿÿÿÿ,_Ò
__sig__2^ôHŠHomologyOfChainComplex„¼çŒ-/usr/local/Magma2.8/package/AlgBas/chainmap.m	0	›
__sig__2rstorsP__sig__0QˆQ Q¸QàQuýÿwûÿûÿûÿûÿÜQdPÐQüv
	__sig__17<dwdPØQûÿÿÿx
IdentityMatrixðyÿÿÿÿÿÿÿÿÔQ„

ZeroModulem0z@True if the map f is a homomorphism of modules over the algebra.d|{w
ûÿûÿûÿûÿôQdPèQ||
	__sig__18œ¼}dPðQûÿÿÿ~IsModuleHomomorphismTÿÿÿÿÿÿÿÿìQ€)/usr/local/Magma2.8/package/AlgBas/attr.m	€	-/usr/local/Magma2.8/package/AlgBas/chainmap.m	€	‚OppositeAlgebra%(%@%…CompactProjectiveResolution$(D(‡CompactInjectiveResolution+´+6ºGiven a projective resolutions P for a simple module S over a basic algebra A, the function returns the chain maps in compact form of the minimal generators for the cohomology Ext*(S,S).8l8Žûÿûÿûÿûÿ@R0R4Rü;0R<Rûÿÿÿ=‘CohomologyRingGenerators>,?D?’ÿÿÿÿÿÿÿÿ8R“"gGiven the generators for cohomolgy, the function returns the list of degrees of the minimal generators.\GH$H”ûÿûÿûÿûÿXR0RLRˆM•
__sig__1N°NøN–0RTRûÿÿÿàPÉæyGiven a module M over a basic algebra and a natural number n the function computes an injective resolution for M out to n steps. The function returns the resolution in compact form together with the coaugmentation map (M -> I_0). The compact form of the resolution is a list of the the minimal pieces of information needed to reconstruct the boundary maps in the resolution. That is the boundary map (I_i -> I_{i-1}) is recorded as a tuple consisting of a matrix whose entries are the images of the generators for indecomposable injective modules making up I_i in the indecomposable injective modules making up I_{i-1} and two lists of integers givin the number of indecomposable projective modules of each isomorphism class in I_i and in I_{i-1}. The actual return of the function is the compact projective resolution of the dual module of M over the opposite algebra of the algebra of M.T$l$Ð8wûÿûÿûÿHSØR<S)Ñ
__sig__4+è+,ÒØRDSûÿÿÿ0.ÓProjectiveResolution01Ôÿÿÿÿÿÿÿÿ@SÕDñThe complex giving the minimal injective resolution of M together with the inclusion homomorphism from M into its injective hull. Note that homomorphisms go from left to right so that the kernel of the first homomorphism in the complex is M.?¨?Ö8wûÿûÿûÿ`SØRTSEØØR\SûÿÿÿHFÙInjectiveResolutionÀHøHÚÿÿÿÿÿÿÿÿXSÛ"gThe nth syzygy module of M. The kernel of the nth boundary map in a minimal projective resolution of M.dQŒQ¤Qg
ismesR§
__sig__4\„¨0RœRûÿÿÿ˜©DegreesOfCohomologyGeneratorsì
ªÿÿÿÿÿÿÿÿ˜R«V7Given the projective resolutions P and Q of two modules M and N and the cohomology generators in compact form C of the cohomology module, Ext_B^*(M,N), the function returns the chain map from P to Q that lifts the nth generator of the of the cohomology ring and has degree equal to the degree of that generator.Ld|¬889ûÿûÿûÿûÿÈR¸R0R¬R|­
__sig__5tŒ®0R´RûÿÿÿT±P$Given the projective resolution P of a module and the cohomology generators in compact form C of the cohomology ring of that module, the function returns the chain map from P to P that lifts the nth generator of the of the cohomology ring and has degree equal to the degree of that generator.+ü+µ,/usr/local/Magma2.8/package/AlgBas/compact.m%	p(	¶,/usr/local/Magma2.8/package/AlgBas/compact.m%	p(	·MThe sequence of sequences of dimensions of the homology groups been computed.:¸ûÿûÿûÿûÿèRØRÜRì<¹
__sig__0>D>l>ºØRäRûÿÿÿÔ?»SimpleHomologyDimensionsBdB¤B¼ÿÿÿÿÿÿÿÿàR½,The sequence of sequences of dimensions of the cohomology groups Ext^j(Si,M) for simple modules Si, to the extent that they have been computed.ÀMNN¾ûÿûÿûÿûÿSØRôRàQøQDRGCTrue if and only if the chain map f is a surjection in every degreeðH9ûÿûÿûÿûÿ(UTUÐJT$UûÿÿÿðKETrue if and only if the chain map f is an isomorphism in every degreeL9ûÿûÿûÿûÿ8UT,U|M
	__sig__15l„NT4Uûÿÿÿìo
__sig__1ulemologyO*ˆTrue if the sequence of chain complexes, 0 -> Domain(f) -> Domain(g) -> Codomain(g) -> 0, where the internal maps are f and g, is exact.TP99ûÿûÿûÿûÿHUT<Uè!Q
	__sig__16`##RTDUûÿÿÿ%SDThe homomorphism induced on homology by the chain map f in degree n.*t*T9uûÿûÿûÿûÿXUTLUü/U
	__sig__17\1t1VTTUûÿÿÿ,6WInducedMapOnHomology8l8XÿÿÿÿÿÿÿÿPUY&wThe connecting homomorphism in degree n of the short exact sequence of chain complexes given by the chain maps f and g.D?\?t?Z99wûÿûÿûÿûÿpUTdUÄD[
	__sig__18ÌEäE\TlUûÿÿÿ$H]ConnectingHomomorphismJ^ÿÿÿÿÿÿÿÿhU`998ûÿûÿûÿûÿˆUT|U Qa
	__sig__19¼RìRi
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|˜UìUûÿÿÿœ}.™Given a homomorphism between two modules over a basic algebra A, returns the dual homomorphism between the dual modules over the opposite algebra O of A.¬Ä~wwûÿûÿûÿûÿÌ\V˜UôUX
__sig__4Xp€˜UüUûÿÿÿ-The injective hull of the nth simple A-module‚ûÿûÿûÿûÿ VV˜UVXƒ
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InjectiveModuleœ"†ÿÿÿÿÿÿÿÿV‡0¦The injective module that is the injective hull of direct sum of S[1] copies of the first simple module for the algebra, S[2] copies of the second simple module, etc.p/ˆûÿûÿûÿûÿ(V˜UV`6‰
__sig__678p8Š˜U$VûÿÿÿP:‹*ˆGiven a module M over a basic algebra A, returns the injective hull I of M as an A-module and the inclusion homomorphism theta: M --> I.?`?x?Œwww8V˜U,VÈD
__sig__7EÐEèEŽ˜U4Vûÿÿÿ(H
InjectiveHullèIÿÿÿÿÿÿÿÿ0V‘)/usr/local/Magma2.8/package/AlgBas/phom.mà7	¨8	’)/usr/local/Magma2.8/package/AlgBas/phom.m(8	¨8	•
__sig__0SðST

IsChainMap$ÿÿÿÿÿÿÿÿHT-True if the chain map f really is a chain map09ûÿûÿûÿûÿhTT\TÀ
__sig__3ØðTdTûÿÿÿ€QThe first and last degree of the domain of f on which the chain map f is defined.x9ûÿûÿûÿxTTlTì
__sig__4!H!œ!TtTûÿÿÿ”#DefinedInDegrees$ü$% ÿÿÿÿÿÿÿÿpT!The module maps of f*ð*"9ÅûÿûÿûÿûÿTT„T¸0#
__sig__51p2À3$TŒTûÿÿÿ6%

ModuleMapsˆ8 8&ÿÿÿÿÿÿÿÿˆT'8ÂThe kernel complex of f and the chain map of the inclusion. In the event that the chain map is not defined on a particular term of the Domain then the entire term is assumed to be in the kernel.üBC(989ûÿûÿûÿÀÇ¨TTœT`G)
__sig__6I I¸I*T¤Tûÿÿÿ´L+OThe cokernel complex of f and the chain map of the projection onto the quotient”P¬PäP,989ûÿûÿûÿ¸TT¬T@T	
ZeroChainMapG`G
ÿÿÿÿÿÿÿÿT1Checks to see if we can take kernel and cokernel.tNœN9ûÿûÿûÿûÿ8TT,TxR
__sig__1ShSSÜûÿûÿûÿûÿxSØRlS	Ý
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__sig__0duleôß$rThe nth injective-syzygy module of M. The cokernel of the nth boundary map in a minimal injective resolution of M.0àûÿûÿûÿûÿˆSØR|SÀá
__sig__7ØðâØR„Sûÿÿÿ€ãInjectiveSyzygyModuleàäÿÿÿÿÿÿÿÿ€Så-/usr/local/Magma2.8/package/AlgBas/creation.m	`H	æ-/usr/local/Magma2.8/package/AlgBas/creation.m	`H	ç&{Given a finite p-group G and a field k of characteristic p and returns the group algebra kG in the form of a basic algebra.ì'(èqæûÿûÿûÿûÿ°S¨S˜SœS-ê˜S¤Sûÿÿÿp0ë&{Given a finite p-group G and a field k of characteristic p and returns the group algebra kG in the form of a basic algebra.h9 9ìqûÿûÿûÿûÿÀS¸S˜S¬S<í
__sig__1=¨=Ð=î˜S´Sûÿÿÿx?ð:ûÿûÿûÿûÿ0WÈS˜S¼SÈDñ
__sig__2EÐEèEò˜SÄSûÿÿÿ(Hó0Creates the tensor product of two basic algebrasKtL´LôûÿûÿûÿûÿØS˜SÌS”Põ
__sig__3Q¼QäQö˜SÔSûÿÿÿPS-
__sig__7\„.T´Tûÿÿÿ˜/>ÜThe image complex I of f and the inclusion of I into the codomain and the projection of the domain onto I. If the chain map has no defined image into a particular term of the codomain then the image is assumed to be zeroÄ09899ûÿûÿì©ÈTT¼TL1
__sig__84T2TÄTûÿÿÿœ3%The sum of the two chain maps f and g¼4999ûÿûÿûÿûÿ¬ÆØTTÌTŒ5
__sig__9Èè6TÔTûÿÿÿØ7/The product of the scalar s and the chain map f"0"h"899ûÿûÿûÿûÿðTèTTÜTÐ&9
	__sig__10\(t(:TäTûÿÿÿ´+;-The composition of the two chain maps f and gl/<999ûÿûÿûÿûÿWøTTìT\6=
	__sig__118l8>TôTûÿÿÿL:?;True if and only if the chain map f is zero in every degree¤<¼<@9ûÿûÿûÿûÿ¬—UTüT@A
	__sig__12üABBTUûÿÿÿECCTrue if and only if the chain map f is an injection in every degree¼HôHD9ûÿûÿûÿûÿUTUNE
	__sig__13€O˜OFTUûÿÿÿ`QI
	__sig__14ŒR¤R:ÈReturns a sequence of all primitive groups of degree d. Some degrees will produce a very large sequence of groups -- in such cases a warning will be printed unless the user specifies Warning := false.xƒæææFûÿûÿûÿûÿZZZL…ZZûÿÿÿ(‡ÿÿÿÿÿÿÿÿZˆ3/usr/local/Magma2.8/package/GrpPerm/aut/backtrack.m¨Q	èX	‰3/usr/local/Magma2.8/package/GrpPerm/aut/extendaut.møQ	èX	Š1/usr/local/Magma2.8/package/GrpPerm/aut/isomgps.mmHR	èX	‹1/usr/local/Magma2.8/package/GrpPerm/aut/isomgps.m˜R	èX	Œ1True if permutation groups G and H are isomorphic@`ææûÿûÿûÿûÿc<Z,Z0ZLŽ
__sig__0¨À,Z8Zûÿÿÿl 0/usr/local/Magma2.8/package/GrpPerm/aut/oddfns.m#øS	èX	‘1/usr/local/Magma2.8/package/GrpPerm/aut/radquot.mHT	èX	—
__sig__0.8/package/§IsInnerism’6/usr/local/Magma2.8/package/GrpPerm/aut/refineseries.m	èX	“1/usr/local/Magma2.8/package/GrpPerm/aut/grpauto.mes.m	èX	”1/usr/local/Magma2.8/package/GrpPerm/aut/grpauto.mˆ8 8•BA finitely presented group F isomorphic to A, and isomorphism F->Aè;<–F"ûÿûÿûÿ`ZPZTZH?˜PZ\Zûÿÿÿ0A™\A finitely presented group O isomorphic to A/Inner(A), and natural epimorphism A`FpGroup->O FHFšF"ûÿûÿûÿpZPZdZôK›
__sig__1M¤MÄMœPZlZûÿÿÿœO
OuterFPGroupQ4QžÿÿÿÿÿÿÿÿhZŸ _Decide whether a is inner and if so also return a corresponding conjugating element of A`Group XPX˜X
PrimitiveGroupsôÿÿÿÿÿÿÿÿ@X%As above, but with a list of degrees.ÀÿûÿûÿûÿûÿhX`X`WTXp
	__sig__12¨À`W\Xûÿÿÿ(RReturns a list of all groups (g) with degree d which satisfy Predicate(g) eq true.8LãÿûÿûÿûÿûÿxXpX`WdX!
	__sig__134"l"`WlXûÿÿÿ<$%As above, but with a list of degrees.'ãÿÿûÿûÿûÿûÿ€X`WtX°,
	__sig__14h.À. `W|XûÿÿÿH1I
__sig__6ocessroup!FReturn the string description of the n-th primitive group of degree d.à9"ûÿûÿûÿûÿX`W„XÀ<#
	__sig__15ø= >$`WŒXûÿÿÿ¨?%PrimitiveGroupDescriptionBB&ÿÿÿÿÿÿÿÿˆX',/usr/local/Magma2.8/package/GrpPerm/trngps.m^	`	( ,/usr/local/Magma2.8/package/GrpPerm/trngps.m^	`	1
__sig__1eGroupProce)HReturns true if the transitive group process has passed its last group. R`RxR*ûÿûÿûÿûÿ°X X¤X`V+
__sig__0WðWXûInternalExtractPrimitiveGroupüÿÿÿÿÿÿÿÿàWý&xReturns the label (s,n) of the primitive group process tuple p. This is the degree and number of the current group of p 0S9ÿ
__sig__7“8Finds the space of projective homomorphisms from M to N.	x

”vûÿûÿûÿûÿXVHVLV”–HVTVûÿÿÿÄ˜ÿÿÿÿÿÿÿÿPV™*/usr/local/Magma2.8/package/AlgBas/stuff.mhb	àh	
__sig__0˜°žhVtVûÿÿÿŸIdempotentGenerators€ ÿÿÿÿÿÿÿÿpV¡
(H¢ûÿûÿûÿûÿVhV„V£
__sig__1ô¤hVŒVûÿÿÿœ!¥NonIdempotentGenerators”#¦ÿÿÿÿÿÿÿÿˆV¨ûÿûÿûÿûÿ@t¨VhVœV¸+©
__sig__2,--ªhV¤Vûÿÿÿp0«
The basis of B¨1¬ûÿûÿûÿûÿŒ—¸VhV¬V 9­
__sig__3:x: :®hV´Vûÿÿÿ<Ã
roductns¯2The index positions of the projective modules of B`?x?°ûÿûÿûÿûÿÈVhV¼VÈD±
__sig__4EÐEèE²hVÄVûÿÿÿ(H³IdempotentPositionsJJ´ÿÿÿÿÿÿÿÿÀVµReturn whether v is a unitÔOìO¶ûÿûÿûÿûÿàVhVÔVPS·__sig__5T˜T`UxUU_:ËThe long exact sequence on homology for the exact sequence of complexes given by the chain maps f and g as a chain complex with the homolgy group in degree i for the Cokernel of C appearing in degree 3i.ÜôbT„UûÿÿÿdcLongExactSequenceOnHomology4dÿÿÿÿÿÿÿÿ€Ue*/usr/local/Magma2.8/package/AlgBas/opalg.mpj	˜p	f*/usr/local/Magma2.8/package/AlgBas/opalg.m¸j	˜p	h>ûÿûÿûÿûÿ¨U˜UœUÀj˜U¤Uûÿÿÿ×,/usr/local/Magma2.8/package/GrpPerm/prmgps.mk	ps	Ø ,/usr/local/Magma2.8/package/GrpPerm/prmgps.mk	ps	
__sig__8imitiveGrouÙGReturns true if the primitive group process has passed its last group. `ˆ°ÚûÿûÿûÿûÿpW`WdWŒÜ`WlWûÿÿÿ¬Ý$InternalPrimitiveGroupProcessIsEmptylÞÿÿÿÿÿÿÿÿhWß<Moves the primitive group process tuple p to its next group ìàûÿûÿûÿûÿûÿˆW`W|Wá
__sig__1  ° â`W„Wûÿÿÿh"ãInternalNextPrimitiveGroupP$h$äÿÿÿÿÿÿÿÿ€Wå _Returns a primitive group process. This will iterate through all groups with degree in Degrees.ü-.,.æÿûÿûÿûÿûÿ°W W`W”Wì5ç
__sig__27d7Œ7è`WœWûÿÿÿÜ9éPrimitiveGroupProcess,;êÿÿÿÿÿÿÿÿ˜Wë&~Returns a primitive group process which will iterate through all groups (g) with degree in Degrees which satisfy Predicate(g).CìãÿÿûÿûÿûÿûÿÀW¸W`W¬W\Gí
__sig__3IœI´Iî`W´Wûÿÿÿ°LïTReturns a primitive group process. This will iterate through all groups of degree d.PàPðûÿûÿûÿûÿÐWÈW`W¼W<T¤RankÒkPathTreeCyclicModulelÿÿÿÿÿÿÿÿ Um,•Given a basic algebra A, creates the opposite algebra. This is the algebra with the same set of elements but with multiplication * given by x*y = yx.¤(nûÿûÿûÿûÿÀU˜U´U¼.p˜U¼UûÿÿÿD1q
OppositeAlgebraÌ4rÿÿÿÿÿÿÿÿ¸Us<ÏGiven a basic algebra A and its opposite algebra O, creates the change of basis matrix B from the vector space of A to the vector space of O, so that if x, y are in A then (xy)B is the same as (yB)(xB) in O.Ô?ì?@twûÿûÿûÿûÿØU˜UÌU„Eu
__sig__2FlF”Fv˜UÔUûÿÿÿtIwBaseChangeMatrixKØKðKxÿÿÿÿÿÿÿÿÐUy&xReturns the dual of a module M over a basic algebra A as a module over the algebra O which is the opposite algebra of A.SìST—PHomeŒU, X¬XûÿÿÿØ2 XÄXûÿÿÿ3InternalNextTransitiveGroup(4ÿÿÿÿÿÿÿÿÀX5 `Returns a transitive group process. This will iterate through all groups with degree in Degrees.ì6ÿûÿûÿûÿûÿðXàX XÔX€7
__sig__2pˆ8 XÜXûÿÿÿð9TransitiveGroupProcess€:ÿÿÿÿÿÿÿÿØX;(Returns a transitive group process which will iterate through all groups (g) with degree in Degrees which satisfy Predicate(g).ì l <ãÿÿûÿûÿûÿûÿYøX XìX<$=
__sig__3%\%t%> XôXûÿÿÿH(?UReturns a transitive group process. This will iterate through all groups of degree d.0.@ûÿûÿûÿûÿYY XüXð5A
__sig__47h77B XYûÿÿÿà9C$pReturns a transitive group process. This will iterate through all groups of degree d which satisfy Predicate(g).=ø= >DãÿûÿûÿûÿûÿY XYBE
__sig__5DPDÈDF XYûÿÿÿFU__sig__8ansitiveGroupLabelGBReturns the current group of the transitive group process tuple p lMŒMHæûÿûÿûÿ(Y XYdQJ X$YûÿÿÿðRKInternalExtractTransitiveGroup`ULÿÿÿÿÿÿÿÿ Yñ
__sig__4¨ò`WÄWûÿÿÿ	ó$oReturns a primitive group process. This will iterate through all groups of degree d which satisfy Predicate(g).$<TôãÿûÿûÿûÿûÿØW`WÌW”õ
__sig__5xö`WÔWûÿÿÿ°÷AReturns the current group of the primitive group process tuple p h€øæûÿûÿûÿèW`WÜW€ù
__sig__6xú`WäWûÿÿÿXþûÿûÿûÿX`WôWì!`WüWûÿÿÿÄ#"InternalExtractPrimitiveGroupLabelL&d&ÿÿÿÿÿÿÿÿøWBReturns the first group (g) of degree d which satisfies Predicate.h.À.ãÿæûÿûÿûÿ XX`WX06`WXûÿÿÿp8QReturns the first group (g) with degree in Degrees which satisfies the predicate.<0<ãÿÿæûÿûÿûÿ0X(X`WXx?	
__sig__9@A0A
`W$Xûÿÿÿ¼C(Returns the first group (g) of degree d.F0FHFæûÿûÿûÿ8X`W,XôK
	__sig__10¤MÄM`W4XûÿÿÿœOûÿûÿûÿûÿXXHX`W<X S
	__sig__11XT€T`WDXûÿÿÿ˜VÐVW¸hVÜVûÿÿÿt¹

Return v^nà	ºûÿûÿûÿûÿàsðVhVäVd»
__sig__6<T¼hVìVûÿÿÿ½
ì¾lllûÿûÿûÿûÿWhVôV€¿
__sig__7pˆÀhVüVûÿÿÿðÁComponentProduct`€ÂÿÿÿÿÿÿÿÿøVÄllûÿûÿûÿûÿÀsWhVWÅ
__sig__8ôÆhVWûÿÿÿœ!bCorootlÇ"Return the minimal polynomial of vü$%Èûÿûÿûÿûÿ(WhVWÀ*É
__sig__9,,°,ÊhV$Wûÿÿÿp/Ë*The field F as a basic algebra over itselfà3ø3Ìqûÿûÿûÿûÿ8WhV,Wà9Í
	__sig__10è:;ÎhV4Wûÿÿÿ`<Ï./usr/local/Magma2.8/package/GrpPerm/dtgroups.m		¹.šGiven a finite permutation group G, with normal subgroup M, if M has a complement in G return true, otherwise false. A single complement is also returned.h¼ÐZìZûÿÿÿ½NGiven a finite permutation group G, with normal subgroups N and M, such that N <= M, this returns a list L of subgroups of G, all satisfying N < L[i] < M, such that L[1]/N and L[i+1]/L[i] are elementary abelian or direct products of nonabelian simple groups for all i, and L[#L] = M$L¾æææûÿûÿûÿûÿ[ÐZôZD¿
__sig__24TÀÐZüZûÿÿÿèÁ
RefineSectionÂÿÿÿÿÿÿÿÿøZÃJGiven a finite permutation group G, with normal subgroups N and M, such that N < M, if M/N has a complement in G/N return a list of representatives of the conjugacy classes of complements in G/N. If M/N does not have a complement in G/N, the empty sequence is returned/ü/Äæææûÿûÿûÿûÿ[ÐZ[Œ6Å
__sig__38„8œ8ÆÐZ[ûÿÿÿœ:ÇJGiven a finite permutation group G, with normal subgroups N and M, such that N < M, if M/N has a complement in G/N return a list of representatives of the conjugacy classes of complements in G/N. If M/N does not have a complement in G/N, the empty sequence is returnedE´EÈæææûÿûÿûÿûÿ4©([ÐZ[JÉ
__sig__4LðLMÊÐZ$[ûÿÿÿ˜NËDòGiven a finite permutation group G, with soluble normal subgroup M, if M has a supplement in G return a list of representatives of the conjugacy classes of supplements in G. If M does not have a supplement in G, the empty sequence is returnedäX,YM&yReturns the label (s,n) of the transitive group process tuple p. This is the degree and number of the current group of p <dNûÿûÿûÿ@Y X4YPO
__sig__7ˆ°P X<Yûÿÿÿ0Q#InternalExtractTransitiveGroupLabel¤¼Rÿÿÿÿÿÿÿÿ8YS&yReturns the first group (g) of degree d which satisfies Predicate, along with a string giving a description of the group.ÄÜTãÿæûÿûÿûÿ`YXY XLYèV XTYûÿÿÿØW*ˆReturns the first group (g) with degree in Degrees which satisfies the predicate, along with a string giving a description of the group.%X%p%XãÿÿæûÿûÿûÿpYhY X\Y´+Y
__sig__9,-Œ-Z XdYûÿÿÿl0[ _Returns the first group (g) of degree d, along with a string giving a description of the group.¼78l8\æûÿûÿûÿxY XlYü;]
	__sig__10Ô<ì<^ XtYûÿÿÿ¤>_:ÉReturns a sequence of all transitive groups of degree d. Some degrees will produce a very large sequence of groups -- in such cases a warning will be printed unless the user specifies Warning := false.¼HôH`ûÿûÿûÿûÿ˜YˆY X|YNa
	__sig__11€O˜Ob X„Yûÿÿÿ`QcTransitiveGroupsR¼RìRdÿÿÿÿÿÿÿÿ€Ye%As above, but with a list of degrees.X-%InternalTransitiveGroupProcessIsEmptyÔR.ÿÿÿÿÿÿÿÿ¨X/=Moves the transitive group process tuple p to its next group d0ûÿûÿûÿûÿûÿÈX X¼XÐ./usr/local/Magma2.8/package/GrpPerm/dtgroups.m	h¡	Û__sig__0pIdentificationÑ$Identify a doubly-transitive group gîÕ4ÒæûÿûÿûÿûÿPW@WDWdÓ
__sig__0¶‰7Ô@WLWûÿÿÿVP9Õ TwoTransitiveGroupIdentification0S9ÖÿÿÿÿÿÿÿÿHWš*/usr/local/Magma2.8/package/AlgBas/stuff.mh¡	h¢	§

entGenerators›
œûÿûÿûÿûÿxVhVlVfÿûÿûÿûÿûÿ¨Y Y X”YHg
	__sig__12Ô
ì
h XœYûÿÿÿ˜iRReturns a list of all groups (g) with degree d which satisfy Predicate(g) eq true.ˆ°jãÿûÿûÿûÿûÿ¸Y°Y X¤YŒk
	__sig__13d|l X¬Yûÿÿÿ¼m%As above, but with a list of degrees.¼nãÿÿûÿûÿûÿûÿÀY X´Y¼o
	__sig__14´Üp X¼YûÿÿÿŒy
PCGroupescriptionqGReturn the string description of the n-th transitive group of degree d.#¨#À#rûÿûÿûÿûÿÐY XÄYD(s
	__sig__15œ*¼*t XÌYûÿÿÿì,uTransitiveGroupDescription40l0vÿÿÿÿÿÿÿÿÈYx//usr/local/Magma2.8/package/GrpPerm/aut/autgp.mœ8w4/usr/local/Magma2.8/package/GrpPerm/aut/attributes.m§	€ª	D8&internal True if the simple Cartan type is simply laceddœEûÿûÿûÿûÿ,]]$\]œF
	__sig__12|”G$\]ûÿÿÿÄH
IsSimplyLacedHIÿÿÿÿÿÿÿÿ]J:&internal True if the compound Cartan type is simply lacedø KÅûÿûÿûÿûÿLc4]$\(]@L
	__sig__13@`M$\0]ûÿÿÿÀN]&internal The maximum multiplicity of an edge in the Dynkin diagram of the simple Cartan typeÜOûÿûÿûÿûÿT]D]$\8]”#P
	__sig__14¤$¼$Q$\@]ûÿÿÿ'RMaximumMultiplicityÀ()Sÿÿÿÿÿÿÿÿ<]T _&internal The maximum multiplicity of an edge in the Dynkin diagram of the compound Cartan typeÐ4˜5ð5UÅûÿûÿûÿûÿ\^\]$\P]P:V
	__sig__15X;€;W$\X]ûÿÿÿÀ<X5&internal The standard injection of the compound typex?YÅÅûÿûÿûÿ|dl]$\`]ÈDZ
	__sig__16ÐEèE[$\h]ûÿÿÿ(H\StandardInjectionJJ]ÿÿÿÿÿÿÿÿd]^D&internal The Cartan matrix for the given type and rank/multiplicityQdQ_ûÿûÿûÿûÿ”]„]$\x]`U`
	__sig__17€V˜Va$\€]ûÿÿÿÐXb
CartanMatrixZ[å
__sig__0(æˆ[”[ûÿÿÿLçMaximalSubgroupsTF(èÿÿÿÿÿÿÿÿ[éFor internal use<Têæûÿûÿûÿûÿ°[ˆ[¤[”ë
__sig__1xìˆ[¬[ûÿÿÿ°í

MaxSubsTF2 Àîÿÿÿÿÿÿÿÿ¨[ïFor internal use(ðæûÿûÿûÿûÿÈ[ˆ[¼[ñ
__sig__2pòˆ[Ä[ûÿÿÿìó

MaxSubsTF3H!œ!ôÿÿÿÿÿÿÿÿÀ[õ2§Find maximal subgroups of G modulo the soluble normal subgroup N. Use MaximalSubgroupsTF on the radical quotient where necessary. A list subgroup records is returned. -è-.öææûÿûÿûÿûÿà[ˆ[Ô[Ð4÷
__sig__36è607øˆ[Ü[ûÿÿÿ 9ùMaximalSubgroupsHè:;úÿÿÿÿÿÿÿÿØ[û2/usr/local/Magma2.8/package/GrpPerm/max/identify.m€µ	P¸	ü2/usr/local/Magma2.8/package/GrpPerm/max/identify.mÐµ	P¸	
__sig__0pleGroupý1Look up G in the database of almost simple groupshH¨Hþææûÿûÿûÿ\\ð[ô[ÄMÿ
__sig__0N4O\Oð[ü[ûÿÿÿ4QIdentifyAlmostSimpleGroupÀRðRÿÿÿÿÿÿÿÿø[1Look up G in the database of almost simple groups¸XÐX__sig__1Z¸Z[@[X[Ï
SupplementsÐÿÿÿÿÿÿÿÿ0[Ñ0¢Given a finite permutation group G, with soluble normal subgroup M, if M has a supplement in G return true, otherwise false. A single supplement is also returned.{//usr/local/Magma2.8/package/GrpPerm/aut/autgp.m(¼	|/The automorphism group of a permutation group GM}æFûÿûÿûÿûÿ¨…üYìYðY~
__sig__0`PìYøYûÿÿÿ€1/usr/local/Magma2.8/package/GrpPerm/aut/autsdgp.màº	(¼	1/usr/local/Magma2.8/package/GrpPerm/aut/autsdgp.m0»	(¼	‚BThe automorphism group of a semidirect product group G=NH fixing NœX„
__sig__0†AutGpSGjæûÿûÿûÿ\ð[\Lð[\ûÿÿÿ(0/usr/local/Magma2.8/package/GrpPerm/max/oddfns.mÀ¼	pÄ	./usr/local/Magma2.8/package/LieTheory/Cartan.m	pÄ		<./usr/local/Magma2.8/package/LieTheory/Cartan.m	pÄ	
)&internal The Cartan type of the string sh€ÅûÿûÿûÿûÿÜ]4\$\(\€$\0\ûÿÿÿð

CartanTypeøÿÿÿÿÿÿÿÿ,\8&internal The name of the simple Cartan type as a stringôûÿûÿûÿûÿ\\L\$\@\Ä#
__sig__1$ü$%$\H\ûÿÿÿ|'

CartanNameÀ()ÿÿÿÿÿÿÿÿD\:&internal The name of the compound Cartan type as a string`1x1Åûÿûÿûÿûÿì]d\$\X\ 8
__sig__2:8:P:$\`\ûÿÿÿÐ;/&internal True if type1 and type2 are isogenousÐ=ø= >ÅÅûÿûÿûÿûÿct\$\h\B
__sig__3DPDÈD$\p\ûÿÿÿF=&internal Prints the Dynkin diagram of the simple Cartan typeJûÿûÿûÿûÿûÿ”\„\$\x\\O 
__sig__4P¬PäP!$\€\ûÿÿÿxR"
DynkinDiagramS#ÿÿÿÿÿÿÿÿ|\$?&internal Prints the Dynkin diagram of the compound Cartan type Z¸Z[Ìææûÿûÿûÿûÿ`[8[ÐZ,[	Í
__sig__5@hÎÐZ4[ûÿÿÿÒæææûÿûÿûÿP[ÐZD[Ó
__sig__64ÔÐZL[ûÿÿÿ`Õ
HasSupplementPÖÿÿÿÿÿÿÿÿH[×NGiven a finite permutation group G, with normal subgroups N and M, such that N < M and M/N is soluble, if M/N has a supplement in G/N return a list of representatives of the conjugacy classes of M/N in G/N. If M/N does not have a supplement in G/N, the empty sequence is returnedl ” ´ Øæææûÿûÿûÿûÿh[ÐZ\[l$Ù
__sig__7%L&d&ÚÐZd[ûÿÿÿx(ÛNGiven a finite permutation group G, with normal subgroups N and M, such that N < M and M/N is soluble, if M/N has a supplement in G/N return a list of representatives of the conjugacy classes of M/N in G/N. If M/N does not have a supplement in G/N, the empty sequence is returned:8:P:Üæææûÿûÿûÿûÿx[ÐZl[ =Ý
__sig__8>ˆ>¨>ÞÐZt[ûÿÿÿ@ß

SupplementBBàÿÿÿÿÿÿÿÿp[á1/usr/local/Magma2.8/package/GrpPerm/max/maxcomp.m°Ê	 Ì	â1/usr/local/Magma2.8/package/GrpPerm/max/maxcomp.mË	 Ì	ã2«G should be a permutation group with trivial soluble radical (i.e. a TF-group). The maximal subgroups of G are computed, including G. A list subgroup records is returned. (T@Täæûÿûÿûÿûÿ˜[ˆ[Œ[ÐX¦NameZ Z GåûÿûÿûÿˆZPZ|Z
¡
__sig__2¨¢PZ„Zûÿÿÿ4£ _Decide whether a is inner and if so also return a corresponding conjugating element of A`Group Èì¤Gåûÿûÿûÿ˜ZPZŒZ€¥
__sig__3pˆ¦PZ”Zûÿÿÿð¨ÿÿÿÿÿÿÿÿZ±
__sig__5ies©HThe characteristic series of A`Group (from top down) used to compute A. XªFûÿûÿûÿûÿ°ZPZ¤Zì!«
__sig__4#d#”#¬PZ¬Zûÿÿÿ%­CharacteristicSeries'|'®ÿÿÿÿÿÿÿÿ¨Z¯BA finitely presented group F isomorphic to A, and isomorphism A->F80p0°FûÿûÿûÿÈZPZ¼Z07²PZÄZûÿÿÿ 9³5/usr/local/Magma2.8/package/GrpPerm/max/complements.m	ÀÔ	´5/usr/local/Magma2.8/package/GrpPerm/max/complements.m	ÀÔ	µBêGiven a finite permutation group G, with normal subgroup M, if M has a complement in G return a list of representatives of the conjugacy classes of complements in G. If M does not have a complement in G, the empty sequence is returned¤MÄM¶ææûÿûÿûÿûÿ[àZÐZÔZ¤Q¸ÐZÜZûÿÿÿ SºæææûÿûÿûÿðZÐZäZxW»
__sig__1XèX0Y‰“ûÿûÿûÿûÿÄ_,^$\ ^H‹$\(^ûÿÿÿ$Œ*True if the matrix C is Cartan irreducibleH`ûÿûÿûÿûÿ<^$\0^tŽ
	__sig__27x$\8^ûÿÿÿ¼IsCartanIrreducibleÄÜ‘ÿÿÿÿÿÿÿÿ4^’/True if the Cartan matrix C is crystallographicì$“ûÿûÿûÿûÿ<cT^$\H^”
	__sig__28lŒ•$\P^ûÿÿÿè–Y&internal The maximum multiplicity of edges in the Dynkin diagram for the Cartan matrix CP$h$—ûÿûÿûÿûÿ”dd^$\X^)˜
	__sig__29ä+ü+™$\`^ûÿÿÿ,.š//usr/local/Magma2.8/package/LieTheory/RootDtm.m@Ý	›ˆ//usr/local/Magma2.8/package/LieTheory/RootDtm.m@Ý	œ
	RootSpaced9œ9ž
CorootSpace´:Ì:Ÿ
SimpleRootsä;ü; 
SimpleCorootsì<¢
CorootSystemÔ¡
CartanMatrix?¤?¥	DimensionÓÔÈÕ£
NumPosRootsàCD©
IsogenyGroupÔVCorootss§FundamentalGroupH,ItI¨
FundamentalHomK«
CoisogenyGroupÔ°SimpleReflectionPermsÕ¬
CoisogenyProjÜª

IsogenyInjxPP±IsCrystallographicßà­
	SimpleReptUŒU®
SchreierVectVW²
IsSemisimpleÔ³
	IsAdjointÛÜ¯BackPointers]p]ˆ]Ð]cÿÿÿÿÿÿÿÿ|]d4&internal The Cartan matrix for a simple Cartan typedeûÿûÿûÿûÿ¤]œ]$\]Pf
	__sig__18ˆ°g$\˜]ûÿÿÿ0h6&internal The Cartan matrix for a compound Cartan typeiÅûÿûÿûÿûÿ´]¬]$\ ]ìj
	__sig__19$<k$\¨]ûÿÿÿ”lThe Cartan matrix of type tn
	__sig__20Ð ü pTrue if C is a Cartan Matrix$¸$mûÿûÿûÿûÿt_¼]$\°]Øo$\¸]ûÿÿÿ˜"qûÿûÿûÿûÿÌ]$\À]t*r
	__sig__21,,,s$\È]ûÿÿÿ¼.t
IsCartanMatrix1uÿÿÿÿÿÿÿÿÄ]v0&internal The Cartan type of the Cartan matrix C9Ä9Ü9wÅûÿûÿûÿûÿ¤_ä]$\Ø]¼<x
	__sig__22ô=>y$\à]ûÿÿÿ¤?zW&internal The list of string names of the irreducible components of the Cartan matrix CDE\E„E{Åûÿûÿûÿûÿ´_ô]$\è]äI|
	__sig__23pL°L}$\ð]ûÿÿÿXN~.Prints the Dynkin diagram of the Cartan matrix0Qûÿûÿûÿûÿûÿ´c^$\ø]|T€
	__sig__24<V\V$\^ûÿÿÿ”X‚,The fundamental group of the Cartan matrix C[Ì[:ReflectionPermutationsØ;ÿÿÿÿÿÿÿÿÜ`<[The permutation giving the action of the rth (co)root of the root datum RD on the (co)roots|”=Øåûÿûÿûÿûÿ<jü`l^ð`È>
	__sig__23¬Ä?l^ø`ûÿÿÿð@ReflectionPermutation Aÿÿÿÿÿÿÿÿô`B5True if the rth root of the root datum RD is positive`CØûÿûÿûÿûÿÌ—al^aLEl^aûÿÿÿ@F5True if the rth root of the root datum RD is negativeì!GØûÿûÿûÿûÿ$al^at%H
	__sig__25ì'(Il^ aûÿÿÿÀ*J

IsNegative,°,KÿÿÿÿÿÿÿÿaL>The index of the negative of the rth root of the root datum RD`6MØûÿûÿûÿûÿ<al^0aÐ:N
	__sig__26è;<Ol^8aûÿÿÿ =P
Negative>ˆ>¨>Qÿÿÿÿÿÿÿÿ4aR The coroots of the root datum RDDPDÈDSØ0ûÿûÿûÿûÿ|iTal^HaøHT
	__sig__27`JxJUl^PaûÿÿÿŒMWÿÿÿÿÿÿÿÿLaX)The positive coroots of the root datum RD`RxRYØ0ûÿûÿûÿûÿŒilal^`a`VZ
	__sig__28ðWX[l^haûÿÿÿ Z\
PositiveCoroots\]ÿÿÿÿÿÿÿÿdaƒ“ûÿûÿûÿûÿ$^^$\^ÄM„
	__sig__254O\O…$\^ûÿÿÿ4QÐ=The simple roots of the root datum RD as the rows of a matrix˜ÑØïÿûÿûÿûÿûÿdhL_l^@_˜Ól^H_ûÿÿÿÀÔ
SimpleRoots¨ÀÕÿÿÿÿÿÿÿÿD_Ö?The simple coroots of the root datum RD as the rows of a matrix4L×Øïÿûÿûÿûÿûÿthd_l^X_LØ
__sig__3\|Ùl^`_ûÿÿÿÜÚ
SimpleCorootsìÛÿÿÿÿÿÿÿÿ\_Ü&The Cartan matrix of the root datum RDèÝØûÿûÿûÿûÿ„h|_l^p_$Þ
__sig__4%(%@%ßl^x_ûÿÿÿ(àThe rank of the root datum RDì*áØûÿûÿûÿûÿ´hŒ_l^€_´0â
__sig__51l2¼3ãl^ˆ_ûÿÿÿŒ6ä"The dimension of the root datum RDÄ9Ü9åØûÿûÿûÿûÿÄhœ_l^_¼<æ
__sig__6=ô=>çl^˜_ûÿÿÿ¤?è$The Cartan type of the root datum RDB¤BéØÅûÿûÿûÿûÿÔh¬_l^ _$Gê
__sig__7H,ItIël^¨_ûÿÿÿðKì0The name of the Cartan type of the root datum RDN0OXOíØûÿûÿûÿûÿäh¼_l^°_ìRî
__sig__8T$T<Tïl^¸_ûÿÿÿ\VðUThe fundamental group of the root datum RD and the projection from the weight latticeœ[ò
__sig__9\ø\ ]†FundamentalGroupSìST‡ÿÿÿÿÿÿÿÿ^ˆ4The fundamental group of the Cartan matrix of type tZœZŠ
	__sig__26ä[\%Åûÿûÿûÿûÿûÿü]œ\$\\$&
__sig__5€˜'$\˜\ûÿÿÿÀ(H&internal The number of positive roots in a root datum of the given typeè)ûÿûÿûÿûÿ¼\¬\$\ \|*
__sig__6l„+$\¨\ûÿÿÿì,
NumPosRoots$<-ÿÿÿÿÿÿÿÿ¤\.H&internal The number of positive roots in a root datum of the given typeT/Åûÿûÿûÿûÿ`Ä\$\¸\è!0
__sig__7#`##1$\À\ûÿÿÿ%2,&internal The dual of the simple Cartan type(t(3ûÿûÿûÿûÿÜ\Ô\$\È\,.4
__sig__80œ0´05$\Ð\ûÿÿÿô36.&internal The dual of the compound Cartan typel87Åûÿûÿûÿûÿ$dä\$\Ø\ü;8
__sig__9<Ô<ì<9$\à\ûÿÿÿ¤>:<&internal True if the simple Cartan type is crystallographicAB;ûÿûÿûÿûÿ]ô\$\è\”F<
	__sig__10¼HôH=$\ð\ûÿÿÿK>IsCrystallographic MÀM?ÿÿÿÿÿÿÿÿì\@>&internal True if the compound Cartan type is crystallographicSAÅûÿûÿûÿûÿL^]$\]tWB
	__sig__11äX,YC$\]ûÿÿÿœ[œïÿ0ûÿûÿxnLnln$žLntnûÿÿÿ<ŸCoxeterMultigraphT ÿÿÿÿÿÿÿÿpn¡CThe FP Coxeter group given by the Coxeter matrix or Cartan matrix M¤¢%"ûÿûÿûÿûÿ˜nnLn„nØ£
__sig__2ì¤LnŒnûÿÿÿœ¥(The FP Coxeter group with isogeny type tÀ¦%"ûÿûÿûÿûÿ¨n nLn”nô§
__sig__3	¤	¨Lnœnûÿÿÿ¼	©'The FP Coxeter group with root datum RDÜ	à	ªØ%"ûÿûÿûÿûÿÈn°nLn¤n
«
__sig__4
(
¬Ln¬nûÿÿÿ@
­3The root datum of the finite Coxeter presentation Gà®"Øûÿûÿûÿûÿ¸pÀnLn´nÌ$¯
__sig__5$à$°Ln¼nûÿÿÿh5±6The Coxeter group of the finite Coxeter presentation GlD²"ØûÿûÿûÿûÿÐnLnÄn˜K³
__sig__6K¬K´LnÌnûÿÿÿRÁ__sig__9tionGroupmäYµUThe labelled directed multigraph of the imprimitive complex reflection group G(p,q,n)¨^¶‹ûÿûÿûÿûÿànLnÔnÜ^·
__sig__7^ð^¸LnÜnûÿÿÿ_¹ImprimitiveReflectionDiagram4gÌRootRestrictionYØYZÍ
IsParabolic¸[Ð[Î
Presentation]t]Ï
StandardGroupü_Ð&The Coxeter group of the root datum RD¼aÑØ;ûÿûÿûÿûÿ\gLgg@gteyÿÿÿÿÿÿÿÿÔezThe constant p_rsØ
ð
{ØûÿûÿûÿûÿôeŒdèeô|
	__sig__15Üô}ŒdðeûÿÿÿÈ~
LieConstant_pÄÿÿÿÿÿÿÿÿìe€SThe length of the left string through the sth root in the direction of the rth rooth€ØûÿûÿûÿûÿfŒdf€‚
	__sig__16xƒŒdfûÿÿÿX„LeftStringLengthô…ÿÿÿÿÿÿÿÿf†The constant q_rsÜ#$‡Øûÿûÿûÿûÿ$fŒdfx(ˆ
	__sig__17Ø*ð*‰Œd fûÿÿÿ-Š
LieConstant_q0‹ÿÿÿÿÿÿÿÿfŒTThe length of the right string through the sth root in the direction of the rth root9:Øûÿûÿûÿûÿ<fŒd0fð<Ž
	__sig__18H>p>Œd8fûÿÿÿØ?RightStringLengthBB‘ÿÿÿÿÿÿÿÿ4f’The constant A_rsG(G“ØûÿûÿûÿûÿTfŒdHfM”
	__sig__19DN\N•ŒdPfûÿÿÿ\P–
LieConstant_A¤Q—ÿÿÿÿÿÿÿÿLf˜,The Cartan integer for the rth and sth rootsV˜V™ØûÿûÿûÿûÿlfŒd`f [š
	__sig__20ü\$]›Œdhfûÿÿÿü_œ
CartanInteger,aÿÿÿÿÿÿÿÿdfžThe constant N_rsüde,eDeJ

CorootNorm„[KÿÿÿÿÿÿÿÿeL1True if the rth root of the root datum RD is longñgÀgûÿÿÿøòStandardParabolicSubgroupóÿÿÿÿÿÿÿÿ¼gô&True if H is a parabolic subgroup of WTõ;;ûÿûÿûÿûÿÜggÐgˆö
__sig__7œ÷gØgûÿÿÿ´øIsStandardParabolicSubgroupÐùÿÿÿÿÿÿÿÿÔgúOThe Coxeter group isomorphic to Coxeter subgroup H, but acting on its own roots¼Àû;;ûÿûÿûÿûÿôggègôýgðgûÿÿÿ¨	þLocalCoxeterGroupÀ	ÿÿÿÿÿÿÿÿÿìg9The root datum whose roots the Coxeter subgroup H acts on
;‹ûÿûÿûÿûÿhgh@

__sig__9
T
ghûÿÿÿä
	Overdatuml!ÿÿÿÿÿÿÿÿh<The Coxeter group whose roots the Coxeter subgroup H acts onl5;;ûÿûÿûÿûÿ$hgh„D
	__sig__10„K	g hûÿÿÿ K

	Overgroup´Kÿÿÿÿÿÿÿÿh%The root datum of the Coxeter group Wˆ^;Øûÿûÿûÿûÿ¸n<hg0h¼^
	__sig__11Ð^g8hûÿÿÿè^%The root space of the Coxeter group W_
	__sig__12ø
IsogenyGroup„ùÿÿÿÿÿÿÿÿÔ_úVThe coisogeny group of the root datum RD and the projection from the fundamental group0ûØ“ûÿûÿûÿ,iô_l^è_Dü
	__sig__11H`ýl^ð_ûÿÿÿŒþ
CoisogenyGroup|ÿÿÿÿÿÿÿÿÿì_The roots of the root datum RDØ0ûÿûÿûÿûÿ<i`l^`ì
	__sig__12ôl^`ûÿÿÿ¼5The number of positive (co)roots of the root datum RD˜!Øûÿûÿûÿûÿ”h`l^`@%
	__sig__13X'x'l^`ûÿÿÿt*
	__sig__15ots,,5The number of positive (co)roots of the root datum RD´0	Øûÿûÿûÿûÿ¤h,`l^ `Œ7

	__sig__14d9œ9l^(`ûÿÿÿü:NumberOfPositiveRoots\<ÿÿÿÿÿÿÿÿ$`'The positive roots of the root datum RD4@A,AØ0ûÿûÿûÿûÿLiD`l^8`äEl^@`ûÿÿÿ$H
PositiveRootsäIÿÿÿÿÿÿÿÿ<`!The rth root of the root datum RDÐOèOØÿÿûÿûÿûÿûÿ\i\`l^P`LS
	__sig__16”T\Ul^X`ûÿÿÿW;The position of the vector v as a root of the root datum RD<[T[
	__sig__17ˆ\°\l^h`ûÿÿÿh_ñØ“ûÿûÿûÿiÌ_l^À_ól^È_ûÿÿÿôSThe isogeny group of the root datum RD and the injection into the fundamental groupõØ“ûÿûÿûÿiÜ_l^Ð_ö
	__sig__10÷l^Ø_ûÿÿÿØûÿûÿûÿûÿlil`l^``H^#The rth coroot of the root datum RD@x_Øÿÿûÿûÿûÿûÿœi„al^xa¨`
	__sig__29ìal^€aûÿÿÿ$cÿÿÿÿÿÿÿÿ|aY
ramotDtmd=The position of the vector v as a coroot of the root datum RDPeØûÿûÿûÿûÿ¬iœal^aPf
	__sig__30`€gl^˜aûÿÿÿàh
CorootPositionðiÿÿÿÿÿÿÿÿ”ax
	__sig__33esricesjBThe reflection matrices of the simple coroots of the root datum RD„"œ"kØ0ûÿûÿûÿûÿìi´al^¨a'l
	__sig__31(¨(ml^°aûÿÿÿ,nSimpleCoreflectionMatricesh.À.oÿÿÿÿÿÿÿÿ¬ap;The reflection matrices of the coroots of the root datum RD8p8qØûÿûÿûÿûÿüiÌal^Àa<r
	__sig__32Ø<ð<sl^Èaûÿÿÿ¨>tCoreflectionMatrices?@uÿÿÿÿÿÿÿÿÄav<The reflection matrix of the rth coroot of the root datum RDG(GwØûÿûÿûÿûÿjäal^ØaMyl^àaûÿÿÿœNzCoreflectionMatrix|P”P{ÿÿÿÿÿÿÿÿÜa|/The height of the rth root of the root datum RDU°UÈU}Øûÿûÿûÿûÿüal^ða Z~
	__sig__34è[\l^øaûÿÿÿD^€

RootHeightt`Œ`
RootPositionhÿÿÿÿÿÿÿÿd`&
	__sig__19trices@The reflection matrices of the simple roots of the root datum RDLdØ0ûÿûÿûÿûÿ¼i„`l^x`à 
	__sig__18è!l^€`ûÿÿÿ€"SimpleReflectionMatrices(#ÿÿÿÿÿÿÿÿ|`$9The reflection matrices of the roots of the root datum RD¨À%ØûÿûÿûÿûÿÌiœ`l^`l"'l^˜`ûÿÿÿ<$£l^ˆbûÿÿÿÔ¤
WeightLattice˜¥ÿÿÿÿÿÿÿÿ„b®
	__sig__42ts˜¦FThe fundamental coweights of the root datum RD as the rows of a matrixð§Øûÿûÿûÿûÿ´j¤bl^˜b,¨
	__sig__41©l^ bûÿÿÿTªFundamentalCoweightsœ«ÿÿÿÿÿÿÿÿœb¬-The lattice of coweights of the root datum RD¼­ØÔûÿûÿûÿûÿÔj¼bl^°b<¯l^¸bûÿÿÿ° °
CoweightLattice"±ÿÿÿÿÿÿÿÿ´b².šThe unique element in the W-orbit of the weight v which lies in the fundamental Weyl chamber, and the word in the generators which sends v to this elementä-ü-³ØuûÿûÿûÿäjÔbl^ÈbÌ4´
	__sig__43ä6,7µl^Ðbûÿÿÿœ9¶
DominantWeightÌ:·ÿÿÿÿÿÿÿÿÌb¸/The orbit of the weight v under the action of WÜ>,?D?¹Ø0ûÿûÿûÿûÿôjìbl^àbDº
	__sig__44œE´E»l^èbûÿÿÿ\G¼
WeightOrbitœI´I½ÿÿÿÿÿÿÿÿäb¾!True if RD1 and RD2 are identical€O˜O¿ØØûÿûÿûÿûÿ˜rcl^øbSÀ
	__sig__45TT|TÁl^cûÿÿÿ”VÂ!True if RD1 and RD2 are isogenousDYŒYÃØØûÿûÿûÿûÿ¨rcl^c@^Ä	__sig__46p`ˆ` `¸`(ReflectionMatricesè&')ÿÿÿÿÿÿÿÿ”`*:The reflection matrix of the rth root of the root datum RD/l/+ØûÿûÿûÿûÿÜi´`l^¨`\6,
	__sig__208l8-l^°`ûÿÿÿL:.ReflectionMatrix;¤;Ì;/ÿÿÿÿÿÿÿÿ¬`D	__sig__24iontions?D?0 `The permutations giving the action of the simple (co)roots on the (co)roots of the root datum RDE,EDE1Ø0ûÿûÿûÿûÿjÌ`l^À`´I2
	__sig__21ØKðK3l^È`ûÿÿÿN4SimpleReflectionPermutationsPXP5ÿÿÿÿÿÿÿÿÄ`6VThe permutations giving the action of the (co)roots of the root datum RD on themselves”V7Øûÿûÿûÿûÿ,jä`l^Ø`œ[8
	__sig__22ø\ ]9l^à`ûÿÿÿø_Ål^cûÿÿÿÔÆ"True if RD1 and RD2 are isomorphic|¬ÇØØûÿûÿûÿûÿdx$cl^c0È
	__sig__47 Él^ cûÿÿÿ°Ê(True if the root datum RD is irreducible¨ÀËØûÿûÿûÿûÿLj4cl^(cTÌ
	__sig__48TlÍl^0cûÿÿÿÎ-True if the root datum RD is crystallographicÏØûÿûÿûÿûÿ\jDcl^8cTÐ
	__sig__49`ØÑl^@cûÿÿÿ,!Ò)True if the root datum RD is simply lacedØ#$ÓØûÿûÿûÿûÿljTcl^Hct(Ô
	__sig__50Ô*ì*Õl^PcûÿÿÿŒ-Ö'True if the root datum RD is semisimple11D1×Øûÿûÿûÿûÿdcl^Xcl8Ø
	__sig__51ô9:Ùl^`cûÿÿÿ|;Ú$True if the root datum RD is adjoint==ÛØûÿûÿûÿûÿxrtcl^hc,AÜ
	__sig__52øBCÝl^pcûÿÿÿ„EÞ
	IsAdjointlF”Fßÿÿÿÿÿÿÿÿlcà-True if the root datum RD is simply connectedÀMáØûÿûÿûÿûÿˆrŒcl^€c Qâ
	__sig__53¼RìRãl^ˆcûÿÿÿ|TäIsSimplyConnected|V”Våÿÿÿÿÿÿÿÿ„cæ"Print the root datum RD at level l8\P\çØûÿûÿûÿûÿûÿ¤cl^˜c(aè
	__sig__54bbÿÿÿÿÿÿÿÿôa‚1The height of the rth coroot of the root datum RD¼ÔƒØûÿûÿûÿûÿbl^bÀ„
	__sig__35,D…l^bûÿÿÿ†
CorootHeightŒ‡ÿÿÿÿÿÿÿÿbˆ<The unique long root of greatest height of the root datum RD¼‰Øÿÿûÿûÿûÿûÿ,bl^ b¼Š
	__sig__36´Ü‹l^(bûÿÿÿŒŒ
HighestLongRootÿÿÿÿÿÿÿÿ$bŽ7The unique root of greatest height of the root datum RD%(%@%ØÿÿûÿûÿûÿûÿDbl^8bì*
	__sig__37Ä,ì,‘l^@bûÿÿÿü/’
HighestRoot\1t1“ÿÿÿÿÿÿÿÿ<b”=The unique short root of greatest height of the root datum RDœ:•Øÿÿûÿûÿûÿûÿ\bl^Pb|=–
	__sig__38¼>Ü>—l^Xbûÿÿÿ4@él^ cûÿÿÿØêHackobjPrintRootDtmôëÿÿÿÿÿÿÿÿœcì-Print the Dynkin diagram of the root datum RD”íØûÿûÿûÿûÿûÿ|j¼cl^°cÈî
	__sig__55¬Äïl^¸cûÿÿÿðð&yThe root datum with whose simple roots are the rows of the matrix A and whose simple coroots are the rows of the matrix B(@ñïÿïÿØûÿûÿûÿûÿÔcÌcl^Àc ò
	__sig__568Lól^Ècûÿÿÿ@ô#The root datum with Cartan matrix C!0!õØûÿûÿûÿûÿäcÜcl^Ðc%ö
	__sig__57ì&'÷l^Øcûÿÿÿ)ø7The irreducible root datum with the given type and rank..0.ùØûÿûÿûÿûÿôcìcl^àcð5ú
	__sig__58h77ûl^ècûÿÿÿà9ü)The root datum with the given simple typeè;<ýØûÿûÿûÿûÿdücl^ðcH?þ
	__sig__59 @8@ÿl^øcûÿÿÿC+The root datum with the given compound typeFFÅØûÿûÿûÿûÿddl^dK
	__sig__60lMŒMl^dûÿÿÿ\O"The root datum with isogeny type tŒQ¤QØûÿûÿûÿûÿ4hdl^dU
	__sig__61ÐVWl^dûÿÿÿ0YThe dual of the rootdatum RD[Ð[	ØØûÿûÿûÿ,dl^ d¼`

	__sig__62¤a¼aIsEqualct˜HighestShortRootE`EˆE™ÿÿÿÿÿÿÿÿTbšDThe fundamental weights of the root datum RD as the rows of a matrixMTM›Øûÿûÿûÿûÿ¤jtbl^hb4Qœ
	__sig__39`RxRl^pbûÿÿÿTžFundamentalWeightsàUVŸÿÿÿÿÿÿÿÿlb +The lattice of weights of the root datum RDè[\¡ØÔûÿûÿûÿûÿÄjŒbl^€bì`¢
	__sig__40Ôaìa-ÿÿÿÿÿÿÿÿ¤d.^The matrix of an inner product which is invariant under the coroot action of the root datum RDl/ØûÿûÿûÿûÿÄdŒd¸dô0
__sig__2`x1ŒdÀdûÿÿÿÄ2
DualCoxeterFormÀ3ÿÿÿÿÿÿÿÿ¼d44The squares of the root lengths of the root datum RDÀ5ØûÿûÿûÿûÿÜdŒdÐdÀ6
__sig__3¸à7ŒdØdûÿÿÿ8
	RootNormsô9ÿÿÿÿÿÿÿÿÔd:6The squares of the coroot lengths of the root datum RD%;ØûÿûÿûÿûÿôdŒdèdÀ*<
__sig__4,,°,=Œdðdûÿÿÿp/>
CorootNorms 1H1?ÿÿÿÿÿÿÿÿìd@=The square of the length of the rth root of the root datum RDP:AØûÿûÿûÿûÿeŒde =B
__sig__5>ˆ>¨>CŒdeûÿÿÿ@D
RootNormABBEÿÿÿÿÿÿÿÿeF=The square of the length of the rth root of the root datum RDxIGØûÿûÿûÿûÿ$eŒde\NH
__sig__6OÔOìOIŒd eûÿÿÿ¤QMØûÿûÿûÿûÿ<eŒd0eUN
__sig__7VÐVWOŒd8eûÿÿÿ0YP

IsLongRoot@[X[Qÿÿÿÿÿÿÿÿ4eR2True if the rth root of the root datum RD is shortDa\aT
__sig__8b4bLbV
IsShortRoot”c¬cl^(dûÿÿÿx.The subroot datum of RD with the given simplesÀ
ØØûÿûÿûÿûÿLd<dl^0dÄ
	__sig__63¬Äl^8dûÿÿÿxSØûÿûÿûÿûÿTeŒdHeHUŒdPeûÿÿÿ$WÿÿÿÿÿÿÿÿLe`
	__sig__10ns X0Set the extraspecial signs for the root datum RDœÄYØûÿûÿûÿûÿûÿleŒd`eLZ
__sig__94T[Œdheûÿÿÿœ\SetExtraspecialSigns<]ÿÿÿÿÿÿÿÿde^AThe index of the sum of the rth and sth root of the root datum RDÈè_Øûÿûÿûÿûÿ„eŒdxe˜!aŒd€eûÿÿÿ#bFThe right string through the sth root in the direction of the rth rootx'cØûÿûÿûÿûÿ”eŒdˆeì,d
	__sig__11/l/eŒdeûÿÿÿt1f
RightString”5ì5gÿÿÿÿÿÿÿÿŒehEThe left string through the sth root in the direction of the rth rootü;iØûÿûÿûÿûÿ¬eŒd eD?j
	__sig__12@4@kŒd¨eûÿÿÿCl

LeftString,EDEmÿÿÿÿÿÿÿÿ¤ev
	__sig__14onabsNIn#The constants epsilon_rs and |N_rs|0MPMoØûÿÄeŒd¸e0Qp
	__sig__13\RtRqŒdÀeûÿÿÿTrLieConstants_epsilonabsNV<V\Vsÿÿÿÿÿÿÿÿ¼etThe constant epsilon_rsœ[´[Ì[uØûÿûÿûÿûÿÜeŒdÐe¸`wŒdØeûÿÿÿèaxLieConstant_epsilonðbxc
RootSubdatumðÿÿÿÿÿÿÿÿ4d)The root subdatum of RD generated by gensTlØ;ûÿûÿûÿûÿTdl^Hd”
	__sig__64¤¼l^Pdûÿÿÿ+The direct sum of the root data RD1 and RD2`ØØØØûÿûÿûÿûÿddl^Xd#
	__sig__65 $¸$l^`dûÿÿÿ' 
	__sig__67tion¤(1The direct sum decomposition of the root datum RD-Œ-Øûÿûÿûÿûÿtdl^hdô3
	__sig__66t6Œ6l^pdûÿÿÿ9LThe standard injection of the adjoint root datum RD into a larger root datum<\<ØØûÿûÿ„dl^xd¤?!l^€dûÿÿÿ¼A"-/usr/local/Magma2.8/package/LieTheory/Const.m	Àv	#8-/usr/local/Magma2.8/package/LieTheory/Const.m	Àv	$NThe maximum multiplicity of an edge in the Dynkin diagram of the root datum RDÀM%ØûÿûÿûÿûÿœdŒdd Q&
__sig__0R¼RìR'Œd˜dûÿÿÿ|T(\The matrix of an inner product which is invariant under the root action of the root datum RDYZ)Øûÿûÿûÿûÿ¬dŒd d8_*
__sig__1` `¸`+Œd¨dûÿÿÿèa,
CoxeterFormÀbØbŸØûÿûÿûÿûÿ„fŒdxfL 
	__sig__21Ø
ð
¡Œd€fûÿÿÿœ¢
LieConstant_N”£ÿÿÿÿÿÿÿÿ|f¤The constant M_rsì¥ØûÿûÿûÿûÿœfŒdf€¦
	__sig__22pˆ§Œd˜fûÿÿÿð¨
LieConstant_M@©ÿÿÿÿÿÿÿÿ”fªThe constant C_ijrs8X«Øûÿûÿûÿûÿ´fŒd¨f¬
	__sig__23!0!­Œd°fûÿÿÿL#®
LieConstant_Cl$¯ÿÿÿÿÿÿÿÿ¬f°The constant eta_rs`*x*±ØûÿûÿûÿûÿÌfŒdÀf0²
	__sig__24`1x1³ŒdÈfûÿÿÿ06´
LieConstant_etaÀ7µÿÿÿÿÿÿÿÿÄf¶QThe structure constants for the integral reductive Lie algebra with root datum RD¨=Ð=·ØûÿûÿûÿûÿäfŒdØfÀA¸
	__sig__25äC D¹ŒdàfûÿÿÿèEºStructureConstantsH(H»ÿÿÿÿÿÿÿÿÜf¼3The reductive Lie algebra over k with root datum RD´NüN½ýÿØÙûÿûÿûÿûÿüfŒdðf¨R¾
	__sig__26ðST¿ŒdøfûÿÿÿVÈ
	RootDatumraW¨WÀ+The semisimple Lie algebra of type t over k@[X[ÁýÿÙûÿûÿûÿûÿgŒdgL`Â
	__sig__27Da\aÃŒdgûÿÿÿ|bÄSemisimpleLieAlgebradädüde[;ûÿûÿûÿûÿèqtighiH]gpiûÿÿÿ$^"The coroots of the Coxeter group W0_;ûÿûÿûÿûÿ˜q„igxiD`
	__sig__31H`ag€iûÿÿÿŒb+The positive coroots of the Coxeter group W4Tc;ûÿûÿûÿûÿ¸q”igˆi|d
	__sig__32t”egiûÿÿÿìf%The rth coroot of the Coxeter group Wèg;ÿÿûÿûÿûÿûÿØq¤ig˜i˜!h
	__sig__33°"H#ig iûÿÿÿ¸$j?The position of the vector v as a coroot of the Coxeter group W¤(¼()k;ûÿûÿûÿûÿøq´ig¨il/l
	__sig__341D1mg°iûÿÿÿì5t
	__sig__36trices7nBThe reflection matrices of the simple roots of the Coxeter group W;,;o;ûÿûÿûÿûÿÄig¸il>p
	__sig__35Œ?¤?qgÀiûÿÿÿ¼Ar;The reflection matrices of the roots of the Coxeter group WÌEäEs;ûÿûÿûÿûÿÔigÈitJugÐiûÿÿÿˆMv<The reflection matrix of the rth root of the Coxeter group WPàPw;ûÿûÿûÿûÿäigØi<Tx
	__sig__37ÜUVygàiûÿÿÿX„
	__sig__40esriceszDThe reflection matrices of the simple coroots of the Coxeter group W^@^{;ûÿûÿûÿûÿôigèib|
	__sig__38ðbxc}gðiûÿÿÿpe~=The reflection matrices of the coroots of the Coxeter group Wøgh(h;ÿÿûÿûÿûÿûÿLhg@hügHhûÿÿÿ¤'The coroot space of the Coxeter group W¨À;ÿÿûÿûÿûÿûÿ\hgPh
	__sig__13ØðgXhûÿÿÿ?The simple roots of the Coxeter group W as the rows of a matrixl„œ;ïÿûÿûÿûÿûÿlhg`hÜ
	__sig__14ÔìghhûÿÿÿLAThe simple coroots of the Coxeter group W as the rows of a matrix ° ;ïÿûÿûÿûÿûÿ|hgphh$
	__sig__15H&`&gxhûÿÿÿt( (The Cartan matrix of the Coxeter group W,Œ,¬,!;ûÿûÿûÿûÿqŒhg€h¤1"
	__sig__166,6#gˆhûÿÿÿl8$3The number of positive roots of the Coxeter group W;,;%;ûÿûÿûÿûÿxqœhghl>&
	__sig__17Œ?¤?'g˜hûÿÿÿ¼A.
	__sig__19otsD(3The number of positive roots of the Coxeter group WüF$G);ûÿûÿûÿûÿhq¬hg hM*
	__sig__18@NXN+g¨hûÿÿÿXP,The rank of the Coxeter group WDR\RtR-;ûÿûÿûÿûÿÈp¼hg°h\V/g¸hûÿÿÿ”X0$The dimension of the Coxeter group W[œ[1;ûÿûÿûÿûÿ¼ƒÌhgÀhˆ`2
	__sig__20paˆa3gÈhûÿÿÿ¨b4&The Cartan type of the Coxeter group Wpe6
	__sig__21Xfpf:
	__sig__22°gÈg;ÿÿûÿûÿûÿûÿDlg8lL
	__sig__69Ø
ð
g@lûÿÿÿœ
WordOnRootSpace”ÿÿÿÿÿÿÿÿ<l4The action of the word w on v in the root space of WÄ;ÿÿûÿûÿûÿûÿ\lgPlX
	__sig__70XpgXlûÿÿÿWordOnCorootSpace@`ÿÿÿÿÿÿÿÿTl+The FP Coxeter group of the Coxeter group Wp;%"ûÿûÿûÿˆntlghl"
	__sig__71¬#Ä#gplûÿÿÿD%&The braid group of the Coxeter group Wx(;"ûÿûÿûÿo„lgxl0. 
	__sig__72 0¸0!g€lûÿÿÿø3"+The pure braid group of the Coxeter group W8p8#;"ûÿûÿûÿo”lgˆl<$
	__sig__73Ø<ð<%glûÿÿÿ¨>&
PureBraidGroupØ?'ÿÿÿÿÿÿÿÿŒl(4The G-set of the Coxeter group W acting on the rootsFHF);îÿûÿûÿûÿûÿ¬lg lôK*
	__sig__74¤MÄM+g¨lûÿÿÿœO,
RootGSetPQ4Q-ÿÿÿÿÿÿÿÿ¤l.6The G-set of the Coxeter group W acting on the coroots`V/;îÿûÿûÿûÿûÿÄlg¸lX[0
	__sig__75Œ\´\1gÀlûÿÿÿl_2

CorootGSetÔ`ì`3ÿÿÿÿÿÿÿÿ¼l4.The Coxeter group W as a real reflection groupte5;jûÿûÿûÿÜlgÐlœjÅÿÿÿÿÿÿÿÿgÆ./usr/local/Magma2.8/package/LieTheory/GrpCox.m	X¦	Çº./usr/local/Magma2.8/package/LieTheory/GrpCox.m	X¦	É
	OverdatumdÊ
	Overgroup ÈÃA"B÷øùúûüýþÒ
__sig__08PÓgHgûÿÿÿˆ5;ÅûÿûÿûÿûÿÜhgÐhH7gØhûÿÿÿ$8 The Cartan type of W as a string09;ûÿûÿûÿûÿqìhgàhD;gèhûÿÿÿ< The Cartan type of W as a string=;ûÿûÿûÿûÿühgðhì>
	__sig__23$<?gøhûÿÿÿ”@
IsogenyType¤¼AÿÿÿÿÿÿÿÿôhB9The fundamental group of the Cartan type of W as a string ° C;ûÿûÿûÿûÿ8qigih$D
	__sig__24H&`&Egiûÿÿÿt(F5The isogeny group of the Cartan type of W as a stringì,G;ûÿûÿûÿûÿHq$igi¼3H
	__sig__25D6\6Ig iûÿÿÿœ8J7The coisogeny group of the Cartan type of W as a string|;¤;Ì;K;ûÿûÿûÿûÿXq4ig(iÜ>L
	__sig__26ì?@Mg0iûÿÿÿ¤BN The roots of the Coxeter group WEœE´EO;0ûÿûÿûÿûÿˆqDig8iJP
	__sig__27ðLMQg@iûÿÿÿ˜NR&Positive roots of the reflection group0QS;0ûÿûÿûÿûÿ¨qTigHi|TT
	__sig__28<V\VUgPiûÿÿÿ”XV#The rth root of the Coxeter group W|[œ[W;ÿÿûÿûÿûÿûÿÈqdigXiˆ`X
	__sig__29paˆaYg`iûÿÿÿ¨bZ=The position of the vector v as a root of the Coxeter group Wf\	__sig__30èfgPgˆgÔ
CoxeterGroup<ÕÿÿÿÿÿÿÿÿDgÖ&The Coxeter group with Cartan matrix CÜ×;ûÿûÿûÿûÿlgdggXgh Ø
__sig__1!°!è!Ùg`gûÿÿÿÀ#ÚThe Coxeter group with type tp%Û;ûÿûÿûÿûÿlltgghg´+Ü
__sig__2,-Œ-Ýgpgûÿÿÿl0Þ0The Coxeter subgroup of W with the given simples56,6ß;;ûÿûÿûÿûÿ”g„ggxgœ:à
__sig__3;¤;Ì;ág€gûÿÿÿì<âReflectionSubgroup„>¤>ãÿÿÿÿÿÿÿÿ|gä<The Coxeter subgroup of W generated by the given reflectionsE„Eå;;ûÿûÿûÿûÿœgggäIæ
__sig__4KpL°Lçg˜gûÿÿÿXNü
__sig__8icSubgroupè$True if H is a Coxeter subgroup of WQDRé;;;ûÿûÿûÿûÿ¬gg gVê
__sig__5WŒW¤Wëg¨gûÿÿÿZìIsReflectionSubgroup[\íÿÿÿÿÿÿÿÿ¤gîCThe standard parabolic subgroup of W generated by the given simples0bHbï;;ûÿûÿûÿûÿÄgg¸gfð
__sig__6fèfgù
	__sig__55\„úpäsûÿÿÿ˜û
Print xüt
Œ
ü,ûÿûÿûÿûÿûÿøspìsý
	__sig__56xþpôsûÿÿÿDÿ
PrintGrpLieElt`ÿÿÿÿÿÿÿÿðs.The Coxeter element of the group of Lie type G¼+,ûÿûÿûÿûÿtptì
	__sig__57ôptûÿÿÿT-The Coxeter number of the group of Lie type G¼+,ûÿûÿûÿûÿ tpth"
	__sig__58Ø#$ptûÿÿÿp%	<Generators for the group of Lie type as an algebraic group G*¼*
+ûÿûÿûÿûÿ0tp$tl0
	__sig__59Œ1¤1p,tûÿÿÿ\6AlgebraicGenerators„8œ8ÿÿÿÿÿÿÿÿ(t&Generators for the group of Lie type G¼<+ûÿûÿûÿûÿìØHtp<t@
	__sig__60üABpDtûÿÿÿEReplace x with x ^ yFDF,,ûÿûÿûÿûÿûÿXtpLtðK
	__sig__61 MÀMpTtûÿÿÿ˜OConjugate of x by yHQ`Q,,,ûÿûÿûÿûÿ¬‰htp\t\U
	__sig__62|V”VpdtûÿÿÿÌXCommutator x^-1 * y^-1 * x * yœ[,,,ûÿûÿûÿûÿxtpltˆ`
	__sig__63paˆapttûÿÿÿ¨b

CommutatorÈdàd ÿÿÿÿÿÿÿÿpt!,/usr/local/Magma2.8/package/LieTheory/Auto.m¹	Pº	",/usr/local/Magma2.8/package/LieTheory/Auto.m¹	Pº	#-The inner automorphism of G induced by x in G„pœpäp0p¼pûÿÿÿ1-The reductive rank of the group of Lie type G2+ûÿûÿûÿûÿÐppÄp4-3
__sig__9d4pÌpûÿÿÿ5-The reductive rank of the group of Lie type G;6+ûÿûÿûÿûÿàppÔpd7
	__sig__10ÿ:8pÜpûÿÿÿ“ t69
ReductiveRank<@:ÿÿÿÿÿÿÿÿØp;.The semisimple rank of the group of Lie type G<+ûÿûÿûÿûÿøppìpf…\9=
	__sig__11€C>pôpûÿÿÿ?
SemisimpleRank@ÿÿÿÿÿÿÿÿðpA,The Cartan matrix of the group of Lie type GLHB+ûÿûÿûÿûÿx°qpqC
	__sig__12˜LDpqûÿÿÿE*The Cartan type of the group of Lie type GdF+ûÿûÿûÿûÿ qpqG
	__sig__13d3 69HpqûÿÿÿI3Print the Dynkin diagram of the group of Lie type G”UJ+ûÿûÿûÿûÿûÿ0qp$qK
	__sig__14Lp,qûÿÿÿàYM0The fundamental group of the group of Lie type GH¹:N+“ûÿûÿûÿûÿ@qp4q*œ6;O
	__sig__15LZPp<qûÿÿÿQ,The isogeny group of the group of Lie type GR+“ûÿûÿûÿûÿPqpDqgS
	__sig__16dTpLqûÿÿÿÄ

WordToPerm\„ÅÿÿÿÿÿÿÿÿkÆ4Converts a word w to a permutation of the roots of WhÇ#;åûÿûÿûÿûÿ$kgkðÈ
	__sig__56\tÉg kûÿÿÿÀÊ;Converts a word w to a matrix acting on the root space of WÄÜË;ïÿûÿûÿûÿûÿDk4kg(kÌ
	__sig__57$Íg0kûÿÿÿ|Î
WordToMatrixŒÏÿÿÿÿÿÿÿÿ,kÐ;Converts a word w to a matrix acting on the root space of W°!è!Ñ#;ïÿûÿûÿûÿûÿLkg@kp%Ò
	__sig__58è'(ÓgHkûÿÿÿ¼*Ô=Converts a word w to a matrix acting on the coroot space of Wl/Õ;ïÿûÿûÿûÿûÿlk\kgPk\6Ö
	__sig__598l8×gXkûÿÿÿL:ØWordToDualMatrix;¤;Ì;ÙÿÿÿÿÿÿÿÿTkÚ=Converts a word w to a matrix acting on the coroot space of W4@Û#;ïÿûÿûÿûÿûÿtkghk´EÜ
	__sig__60üF$GÝgpkûÿÿÿ´IÞ'Converts a permutation p in W to a wordPMhMˆMßå;ûÿûÿûÿûÿ„kgxk`Qà
	__sig__61ŒR¤Rág€kûÿÿÿ<Tâ

PermToWordÜUVãÿÿÿÿÿÿÿÿ|kä;Converts a matrix M acting on the root space of W to a wordˆ\°\åïÿ;ûÿûÿûÿûÿœkgkXaæ
	__sig__620bHbçg˜kûÿÿÿ°dè
MatrixToWorde°eéÿÿÿÿÿÿÿÿ”k_
Function;ûÿûÿûÿûÿjgøi€
	__sig__39gjûÿÿÿ‚>The reflection matrix of the rth coroot of the Coxeter group Wƒ;ûÿûÿûÿûÿjgj…gjûÿÿÿ”	__sig__44iontions†BThe action of the simple roots on the roots of the Coxeter group W‡;ûÿûÿûÿûÿ$jgjˆ
	__sig__41‰g jûÿÿÿŠ;The action of the roots on the roots of the Coxeter group W‹;ûÿûÿûÿûÿ4jg(jŒ
	__sig__42g0jûÿÿÿŽ>The action of the rth root on the roots of the Coxeter group W;åûÿûÿûÿûÿDjg8j
	__sig__43‘g@jûÿÿÿ’)True if W is an irreducible Coxeter group“;ûÿûÿûÿûÿXrTjgHj•gPjûÿÿÿ–-True if W is a crystallographic Coxeter group—;ûÿûÿûÿûÿdjgXj˜
	__sig__45™g`jûÿÿÿšHTrue if the underlying root datum of the Coxeter group W is simply laced›;ûÿûÿûÿûÿhrtjghjœ
	__sig__46gpjûÿÿÿž/Print the Dynkin diagram of the Coxeter group WŸ;ûÿûÿûÿûÿûÿ(q„jgxj 
	__sig__47¡g€jûÿÿÿ¢EThe degrees of the basic polynomial invariants of the Coxeter group W£;ûÿûÿûÿûÿ”jgˆj¤
	__sig__48¥gjûÿÿÿ¦
BasicDegrees§ÿÿÿÿÿÿÿÿŒj¨FThe fundamental weights of the Coxeter group W as the rows of a matrix©;ûÿûÿûÿûÿ(r¬jg jª
	__sig__49«g¨jûÿÿÿ²
	__sig__51ts¬HThe fundamental coweights of the Coxeter group W as the rows of a matrix­;ûÿûÿûÿûÿ8r¼jg°j®
	__sig__50¯g¸jûÿÿÿ°)The weight lattice of the Coxeter group W±;ûÿûÿûÿûÿrÌjgÀj³gÈjûÿÿÿ´)The weight lattice of the Coxeter group Wµ;ûÿûÿûÿûÿrÜjgÐj¶
	__sig__52·gØjûÿÿÿ¸GThe dominant weight of the Coxeter group W in the orbit of the weight v¹;ÿÿûÿûÿûÿûÿHrìjgàjº
	__sig__53»gèjûÿÿÿ¼0The orbit the weight v under the Coxeter group W½;ÿÿûÿûÿûÿûÿüjgðj¾
	__sig__54¿gøjûÿÿÿÀ4Converts a word w to a permutation of the roots of WÁ;åûÿûÿûÿûÿkkgkÂ
	__sig__55Ãgkûÿÿÿê=Converts a matrix M acting on the coroot space of W to a word˜ëïÿ;ûÿûÿûÿûÿ´kg¨k˜ì
	__sig__63xíg°kûÿÿÿÀîDualMatrixToWord\tïÿÿÿÿÿÿÿÿ¬kðOConverts a matrix M acting on the root space of W to a permutation on the roots¼äüñïÿ;åûÿûÿûÿûÿÌkgÀk$ò
	__sig__64$DógÈkûÿÿÿŒog¸mûÿÿÿØp
RightDescentSetœqÿÿÿÿÿÿÿÿ´mrYThe additive ordering on roots of the Coxeter group W corresponding to the reduced word w|”s;ûÿûÿûÿûÿämÔmgÈmÄt
	__sig__87¨ÀugÐmûÿÿÿàv
AdditiveOrderwÿÿÿÿÿÿÿÿÌmx5An additive order of the roots of the Coxeter group WHy;ûÿûÿûÿûÿpoìmgàmz
	__sig__88ô{gèmûÿÿÿœ!|7The group of the standard action of the Coxeter group Wl$¤$¼$};æûÿûÿûÿümgðmx*~
	__sig__89,0,gømûÿÿÿÀ.€StandardActionGroup 1H1ÿÿÿÿÿÿÿÿôm‚*The standard action of the Coxeter group WÈ9à9ƒ;æ0ûÿûÿngnÀ<„
	__sig__90ø= >…gnûÿÿÿ¨?†
StandardActionhA‡ÿÿÿÿÿÿÿÿnˆ(The transversal element in the coset H xG@G`G‰å;;åûÿûÿûÿûÿ,ng nTMŠ
	__sig__91tNœN‹g(nûÿÿÿ”PŒ
TransversalEltäQÿÿÿÿÿÿÿÿ$nŽ"lThe indexed set of coset representatives of shortest length of the Coxeter subgroup H in the Coxeter group WYZ;;0ûÿûÿûÿûÿDng8n<_
	__sig__92¤`¼`‘g@nûÿÿÿìa’//usr/local/Magma2.8/package/LieTheory/CoxPres.mpì	“0//usr/local/Magma2.8/package/LieTheory/CoxPres.mpì	”CoxeterMultigraphh,h–
CoxeterMatrixìkô
MatrixToPermõÿÿÿÿÿÿÿÿÄköQConverts a matrix M acting on the coroot space of W to a permutation on the roots¤$¼$÷ïÿ;åûÿûÿûÿûÿäkgØkx*ø
	__sig__65,0,ùgàkûÿÿÿÀ.úDualMatrixToPerm1 1H1ûÿÿÿÿÿÿÿÿÜküMConverts a permutation of the roots of W to a matrix acting on the root spaceÐ:ýå;ïÿûÿûÿûÿûÿükgðkÐ=þ
	__sig__660?H?ÿgøkûÿÿÿ0AºÿÿÿÿÿÿÿÿØn»NThe standard presentation of the imprimitive complex reflection group G(p,q,n)¤¼%"ûÿûÿûÿûÿønLnìn½
__sig__8ü¾Lnônûÿÿÿ`¿-The braid group of the Coxeter presentation WÀ""ûÿûÿûÿûÿoLnünìÂLnoûÿÿÿ|Ã2The pure braid group of the Coxeter presentation W¤¼Ä""ûÿûÿûÿûÿoLno<Å
	__sig__10 h ÆLnoûÿÿÿ"Ç9The action of the Coxeter presentation W on the (co)roots(%@%È"ûÿûÿûÿûÿ(oLnoì*É
	__sig__11Ä,ì,ÊLn$oûÿÿÿü/Ë:The action of the Coxeter presentation W on the root space6,6Ì"ûÿûÿûÿûÿ8oLn,oœ:Í
	__sig__12¤;Ì;ÎLn4oûÿÿÿì<Ï<The action of the Coxeter presentation W on the coroot space?Ô?Ð"ûÿûÿûÿûÿHoLn<oDEÑ
	__sig__13,FDFÒLnDoûÿÿÿôHÓ1The longest element in the Coxeter presentation W0MPMÔ"#ûÿûÿûÿûÿXoLnLo0QÕ
	__sig__14\RtRÖLnToûÿÿÿT×1The Coxeter element in the Coxeter presentation WTWtWØ"#ûÿûÿûÿûÿthoLn\o\Ù
	__sig__15ˆ]Ð]ÚLndoûÿÿÿˆ`Û/A additive order for the Coxeter presentation WHb`bxbÜ"#ûÿûÿûÿûÿxoLnlo@fÝ
	__sig__16PgˆgÞLntoûÿÿÿˆk
PermToMatrixCDÿÿÿÿÿÿÿÿôkMConverts a permutation of the roots of W to a matrix acting on the root spaceðKå;ïÿûÿûÿûÿûÿlglXP
	__sig__67ˆQ QglûÿÿÿSPermToDualMatrixT”T\Uÿÿÿÿÿÿÿÿl?The action of the word w on the rth root of the Coxeter group WÌ[ä[\	;ûÿûÿûÿûÿ,lg lè`

	__sig__68Ðaèag(lûÿÿÿxc

WordOnRoot(e@eÿÿÿÿÿÿÿÿ$l4The action of the word w on v in the root space of Wkèkß8Returns the reduced word of s_r*w, assuming w is reduced`˜à"#ûÿûÿûÿûÿˆoLn|o˜á
	__sig__17xâLn„oûÿÿÿÀãSimpleLeftProduct\täÿÿÿÿÿÿÿÿ€oå8Returns the reduced word of w*s_r, assuming w is reduced4Tæ"#ûÿûÿûÿûÿ oLn”o|ç
	__sig__18t”èLnœoûÿÿÿìéSimpleRightProduct4Lêÿÿÿÿÿÿÿÿ˜oëAReturns the reduced word of w1*w2, assuming they are both reduced°"H#ì"#ûÿûÿûÿûÿ¸oLn¬ox'í
	__sig__19¼()îLn´oûÿÿÿ,,ï
WordProduct.,.ðÿÿÿÿÿÿÿÿ°oñA reduced expression for wD6\6òûÿ"ûÿûÿûÿûÿûÿØoÐoLnÄoÌ:ó
	__sig__20ä;ü;ôLnÌoûÿÿÿ=õA reduced expression for w,?D?ö"#ûÿûÿûÿûÿpàoLnÔoD÷
	__sig__21œE´EøLnÜoûÿÿÿ\GùThe inverse of wIÌIäIú"#ûÿûÿûÿûÿ°sðoLnäoøNû
	__sig__22xPPüLnìoûÿÿÿDRýTrue if w1 equals w2SŒSþ"ûÿûÿûÿûÿpLnôoXÿ
	__sig__23ÔYZLnüoûÿÿÿ\ÿÿÿÿÿÿÿÿøo./usr/local/Magma2.8/package/LieTheory/GrpLie.m	`		~./usr/local/Magma2.8/package/LieTheory/GrpLie.m	`		7The group of Lie type with Weyl group W over the ring k f¸fÐfýÿ;+ûÿûÿûÿûÿ0p ppp˜l°lÈl6
	__sig__76ü$7gØlûÿÿÿH80The Coxeter group W as a real coreflection grouph9;jûÿûÿûÿìlgàlð:
	__sig__77\t;gèlûÿÿÿÀU.The coisogeny group of the group of Lie type G¤V+“ûÿûÿûÿûÿ`qpTqÔW
	__sig__170Xp\qûÿÿÿP_
	__sig__19ots°Y7The number of positive roots of the group of Lie type Gð,Z+ûÿûÿûÿûÿpqpdq¼[
	__sig__18Ôì\plqûÿÿÿ|]7The number of positive roots of the group of Lie type G¼Ôì^+ûÿûÿûÿûÿ€qptq`p|qûÿÿÿü a$The roots of the group of Lie type G##b+0ûÿûÿûÿûÿqp„q(c
	__sig__20\*t*dpŒqûÿÿÿ¬,e&The coroots of the group of Lie type Gl0f+0ûÿûÿûÿûÿ qp”q,7g
	__sig__21ä89hpœqûÿÿÿÌ:i-The positive roots of the group of Lie type G¼<j+0ûÿûÿûÿûÿ°qp¤q@k
	__sig__22üABlp¬qûÿÿÿEm/The positive coroots of the group of Lie type G\GH$Hn+0ûÿûÿûÿûÿÀqp´qˆMo
	__sig__23°NøNpp¼qûÿÿÿàPq'The rth root of the group of Lie type GìRSSr+ÿÿûÿûÿûÿûÿpºÐqpÄqtWs
	__sig__24äX,YtpÌqûÿÿÿœ[u)The rth coroot of the group of Lie type G _8_v+ÿÿûÿûÿûÿûÿàqpÔqHbw
	__sig__25c¨cxpÜqûÿÿÿ°ey6The position of v as a root of the group of Lie type Gøgz+ûÿûÿûÿûÿðqpäqxmãEltList|n¹FûÿûÿûÿûÿìvÀvàv$¼AutomorphismGroupPGroup2@½ÿÿÿÿÿÿÿÿäv¾//usr/local/Magma2.8/package/GrpPC/pgrps/autos.m0		¿3/usr/local/Magma2.8/package/GrpPC/pgrps/canonical.m(		0		À3/usr/local/Magma2.8/package/GrpPC/pgrps/canonical.mx		0		ÁV:G is a unipotent subgroup of GL (d, F), U is subspace of natural vector space, determine stabiliser in G of U; if ComputeBase is false, we assume that the generators of G define a decreasing chief series for G; return stabiliser of U, canonical element C of orbit of U under G, and element x of G such that U^x = C´	Âtjjtkûÿûÿwwwè	Ã
__sig__0	ü	Äwwûÿÿÿ
ÅUnipotentStabiliser,
ÆÿÿÿÿÿÿÿÿwÇ2/usr/local/Magma2.8/package/GrpPC/pgrps/centrals.mP		0		È,/usr/local/Magma2.8/package/GrpPC/pgrps/cf.m		0		g$(/usr/local/Magma2.8/package/Plane/geom.mpecÏ		ä»çÉ5/usr/local/Magma2.8/package/GrpPC/pgrps/char-spaces.m		0		Ë./usr/local/Magma2.8/package/GrpPC/pgrps/misc.m	0		Ì./usr/local/Magma2.8/package/GrpPC/pgrps/misc.m	0		Í$tThe p-covering group of H; if Exponent is non-zero, then enforce the corresponding exponent law for the p-cover of H|^Î*ûÿûÿ@w0w4w°^Ï
__sig__0^Ä^Ð0w<wûÿÿÿÜ^ÑBconstruct subspace for allowable subgroup of P whose quotient is G_ÒtûÿûÿPw0wDw´vœ[•
ÿÔ÷—:The presentation of the given labelled directed multigrapha(a˜0"ûÿûÿûÿûÿhnLn\nàd™
__sig__0eÈeàešLndnûÿÿÿg›&The multigraph of a symmetric matrix C¸k
__sig__1l°lÈlMg0mûÿÿÿÜN.The Coxeter length of w in the Coxeter group WìOå;ûÿûÿûÿûÿDmg8mP
	__sig__81Qg@mûÿÿÿ€R
CoxeterLength<Sÿÿÿÿÿÿÿÿ<mTThe longest element of W#`##U;åûÿûÿûÿûÿPo\mgPm(V
	__sig__82\*t*WgXmûÿÿÿ¬,X
LongestElement¼.YÿÿÿÿÿÿÿÿTmZThe Coxeter element of W6t6Œ6[;ÿÿûÿûÿûÿûÿ`otmghmü:\
	__sig__83<,<]gpmûÿÿÿ|=^
CoxeterElementÜ>_ÿÿÿÿÿÿÿÿlm`The Coxeter number of WDLDÄDa;ÿÿûÿûÿûÿûÿtŒmg€môHb
	__sig__84\JtJcgˆmûÿÿÿˆMd
CoxeterNumberøNeÿÿÿÿÿÿÿÿ„mfIThe set of indices r such that the length of s_r*w is less than that of w”T\Ugå;ûÿûÿûÿûÿ¤mg˜mŒYh
	__sig__85|[œ[ig mûÿÿÿp]j
LeftDescentSetø_kÿÿÿÿÿÿÿÿœmlIThe set of indices r such that the length of w*s_r is less than that of wXepemå;ûÿûÿûÿûÿ¼mg°m˜jn
	__sig__86llÅBruhatint<CoreflectionGroup=ÿÿÿÿÿÿÿÿäl>!The action of W on the root space?;ûÿûÿûÿûÿ0omgøl@
	__sig__78AgmûÿÿÿB

RootActionCÿÿÿÿÿÿÿÿülD#The action of W on the coroot spaceE;ûÿûÿûÿûÿ@omgmF
	__sig__79GgmûÿÿÿH
CorootActionIÿÿÿÿÿÿÿÿmJ.The Coxeter length of w in the Coxeter group WKå;ûÿûÿûÿûÿ4mg(mL
	__sig__80{
	__sig__26\„|pìqûÿÿÿ˜}8The position of v as a coroot of the group of Lie type G¼Ô~+ûÿûÿûÿûÿrpôqÀ
	__sig__27,D€püqûÿÿÿ-The weight lattice of the group of Lie type GL‚+ØûÿûÿûÿûÿrprLƒ
	__sig__28\|„prûÿÿÿÜ…/The coweight lattice of the group of Lie type GÜô†+Øûÿûÿûÿûÿ rpr° ‡
	__sig__29""ˆprûÿÿÿ$‰2The fundamental weights of the group of Lie type GX'x'Š+Øûÿûÿûÿûÿ0rp$rì,‹
	__sig__30/l/Œp,rûÿÿÿt1“
	__sig__32ts5ì52The fundamental weights of the group of Lie type Gd9œ9Ž+Øûÿûÿûÿûÿ@rp4rŒ<
	__sig__31¤=Ì=p<rûÿÿÿt?‘.The dominant weight of the group of Lie type G¤B’+ØûÿûÿûÿûÿPrpDr$G”pLrûÿÿÿ´I•.True if the group of Lie type G is irreducibleˆM–+Øûÿûÿûÿûÿ\ž`rpTr`Q—
	__sig__33ŒR¤R˜p\rûÿÿÿ<T™/True if the group of Lie type G is simply lacedtWŒW¤Wš+ØûÿûÿûÿûÿprpdrP\›
	__sig__34^@^œplrûÿÿÿ¸`*True if the group of Lie type G is adjoint`bxbž+Øûÿûÿûÿûÿ€rptr@fŸ
	__sig__35Pgˆg p|rûÿÿÿˆk¡3True if the group of Lie type G is simply connectedm¨m£	__sig__36¤o¼op$p
__sig__0ü$ppûÿÿÿH	
GroupOfLieTypeì

ÿÿÿÿÿÿÿÿp:The group of Lie type with Cartan matrix C over the ring k`ýÿ+ûÿûÿûÿûÿ@p8pp,p\
__sig__14Lp4pûÿÿÿ„8The group of Lie type with root datum RD over the ring k$<ýÿØ+ûÿûÿûÿûÿPpHpp<p
__sig__24LpDpûÿÿÿ<7The group of Lie type of isogeny type t over the ring kè!""ýÿ+ûÿûÿûÿûÿXppLp`&
__sig__3($(D(pTpûÿÿÿì*+True if G automatically normalises elementsd.¼.+ûÿûÿûÿûÿhpp\p,6
__sig__47¤7¼7pdpûÿÿÿ:¢+Øûÿûÿûÿûÿrp„rL¤pŒrûÿÿÿ(¥True if G and H are the same¦++ûÿûÿûÿûÿÐs rp”r§
	__sig__374¨pœrûÿÿÿ`©True if G and H are isogenous°ª++ûÿûÿûÿûÿ°rp¤rÀ«
	__sig__38Øð¬p¬rûÿÿÿHµ
	__sig__40morphic­2True is G and H are isomorphic as algebraic groups@®++ûÿûÿûÿûÿÀrp´rœ"¯
	__sig__39$$<$°p¼rûÿÿÿd&±IsAlgebraicallyIsomorphic(¨(²ÿÿÿÿÿÿÿÿ¸r³=Internal for printing the group of Lie type G in Magma formatH1´+ûÿûÿûÿûÿûÿØrpÌrp8¶pÔrûÿÿÿP:·InternalMagmaPrint¨;Ð;¸ÿÿÿÿÿÿÿÿÐrº2A random element of the finite group of Lie type Gð?@¹+,ûÿûÿûÿûÿˆƒìrpèrˆE¼
	__sig__43pF˜F»pðrûÿÿÿxI½/The identity element of the group of Lie type GTMlMŒM¾+,ûÿûÿûÿûÿˆspôrdQ¿
	__sig__42R¨RÀpürûÿÿÿ@TÁThe bruhat decomposition of g˜VÂ,,,,,ûÿsps [Ã
	__sig__44ü\$]Äpsûÿÿÿü_ÆÿÿÿÿÿÿÿÿsÇ5True if x is normalised (in the Bruhat decomposition)ädÈ,ûÿûÿûÿûÿ(spsÌgÉ
	__sig__45dkŒkÊp$sûÿÿÿÌlË
IsNormalisedmÜmÍNormalise xp(pppˆp
IsNormalising<ÿÿÿÿÿÿÿÿ`pSet the Normalising flag??¨?+ûÿûÿûÿûÿûÿ€pptpE
__sig__5EFF p|pûÿÿÿ¨H!
SetNormalisingJ"ÿÿÿÿÿÿÿÿxp#)The Weyl group of the group of Lie type G|P”P$+;ûÿûÿûÿûÿ˜ppŒpT%
__sig__6U°UÈU&p”pûÿÿÿ¨W'
	WeylGroupHYY(ÿÿÿÿÿÿÿÿp)(The base ring of the group of Lie type G_4`L`*+ýÿûÿûÿûÿûÿüƒ°pp¤pÜb+
__sig__7düde,p¬pûÿÿÿDf-)The root datum of the group of Lie type Giœj.+ØûÿûÿûÿûÿÀpp´pÜm/
__sig__8op(pí&ûÿûÿûÿûÿ¼{”{°{Hï”{¸{ûÿÿÿ$ð\The sequence of elliptic curves with label S (e.g. "101A") in the elliptic curve database D.ñ5ûÿûÿûÿûÿÔ{Ì{”{À{Àò
__sig__2¤¼ó”{È{ûÿÿÿÜô"gThe sequence of elliptic curves of conductor N and label S (e.g. "A") in the elliptic curve database D.¼Ôìõ5ûÿûÿûÿûÿÜ{”{Ð{Üö
__sig__3T÷”{Ø{ûÿÿÿøYAn elliptic curve with label S (e.g. "101A" or "101A1") in the elliptic curve database D. $8$ù5ûÿûÿûÿûÿô{ì{”{à{¤(ú
__sig__4*+´+û”{è{ûÿÿÿü-ü _The J-th elliptic curve of conductor N and label S (e.g. "A") in the elliptic curve database D.,6D6\6ý5ûÿûÿûÿûÿè€ü{”{ð{Ì:þ
__sig__5;ä;ü;ÿ”{ø{ûÿÿÿ=>The number of curves isogeny class defined by S (e.g. "101A").@5ûÿûÿûÿûÿ||”{|„E
__sig__6FlF”F”{|ûÿÿÿtICThe number of curves of conductor N and isogeny class S (e.g. "A").NN5ûÿûÿûÿûÿ|”{|àQ
__sig__7RSS”{|ûÿÿÿ\U//usr/local/Magma2.8/package/CrvEll/ec_ff_misc.mS			//usr/local/Magma2.8/package/CrvEll/ec_ff_misc.mS		
-The trace of the Frobenius endomorphism of E.8_&ûÿûÿûÿûÿT|4|$|(|Hb
__sig__0cc¨c$|0|ûÿÿÿ°eTraceOfFrobeniusf¸fÐfÿÿÿÿÿÿÿÿ,|[The trace of the r-th power Frobenius endomorphism, where E is defined over a finite field.¤o¼o&ûÿûÿûÿûÿd|L|$|@|tu__sig__1v\v€v˜vØvt þÿgûÿûÿûÿûÿàØ¨Ì$w /The hyperelliptic curve given by y^2 + h y = f.HLx þÿgûÿûÿûÿûÿðè¨Ü€y 
__sig__2”z ¨äûÿÿÿ¬{ )The hyperelliptic curve given by y^2 = f.Ð| gûÿûÿûÿûÿX‚ø¨ìœ} 
__sig__3°~ ¨ôûÿÿÿÈ— 
__sig__9eOfGenus :The hyperelliptic curve of genus g given by y^2 + h y = f.¤	€ gûÿûÿûÿûÿ‚‚¨üØ	 
__sig__4	ì	‚ ¨‚ûÿÿÿ
ƒ HyperellipticCurveOfGenus 
„ ÿÿÿÿÿÿÿÿ‚… :The hyperelliptic curve of genus g given by y^2 + h y = f.ä† ÿgûÿûÿûÿûÿ(‚ ‚¨‚Ð$‡ 
__sig__5$ä$ˆ ¨‚ûÿÿÿl5‰ :The hyperelliptic curve of genus g given by y^2 + h y = f.tDŠ þÿgûÿûÿûÿûÿ8‚0‚¨$‚ K‹ 
__sig__6K´KŒ ¨,‚ûÿÿÿR :The hyperelliptic curve of genus g given by y^2 + h y = f.Œ^Ž þÿgûÿûÿûÿûÿH‚@‚¨4‚À^ 
__sig__7^Ô^ ¨<‚ûÿÿÿì^‘ 4The hyperelliptic curve of genus g given by y^2 = f.$g’ gûÿûÿûÿûÿP‚¨D‚¼v!
tRing«@zÏ'

Stabiliser/package/±AutomorphismGroup$,+ûÿûÿûÿûÿ˜tˆtŒtì
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__sig__0h&ˆt”tûÿÿÿ'InnerAutomorphism¨À(ÿÿÿÿÿÿÿÿt1
__sig__2ismð)6The diagonal automorphism of G induced by the vector vÜ*+ûÿûÿûÿûÿ°tˆt¤t+
__sig__1$,ˆt¬tûÿÿÿ|-DiagonalAutomorphismÜ.ÿÿÿÿÿÿÿÿ¨t/IThe graph automorphism of G induced by p, a permutation of degree Rank(G)€"˜"0å+ûÿûÿûÿûÿÈtˆt¼t'2ˆtÄtûÿÿÿ)3GraphAutomorphism,,,4ÿÿÿÿÿÿÿÿÀt5CThe graph automorphism of G induced by the field automorphism sigmaD6\66ûÿûÿûÿûÿàtˆtÔtÌ:7
__sig__3;ä;ü;8ˆtÜtûÿÿÿ=9FieldAutomorphism¼>Ü>:ÿÿÿÿÿÿÿÿØt;,/usr/local/Magma2.8/package/LieTheory/Repn.mb		ˆf		< ,/usr/local/Magma2.8/package/LieTheory/Repn.mb		ˆf		=&}List of triples giving the position and value of the non-zero entries of the matrix of ad(e_alpha) acting on X (on the right)XO>ØûÿûÿûÿûÿuðtôtìR?
__sig__0T$T<T@ðtütûÿÿÿ\VA

LieRootEltìWXBÿÿÿÿÿÿÿÿøtC*ˆThe matrix of ad(e_alpha) acting on X (on the right), where X is an sequence of positive root indices or of negative root indices for RDaÐaèaDØûÿûÿûÿûÿuðtu°eE
__sig__1fˆf fFðtuûÿÿÿ(hG
LieRootMatrixèkHÿÿÿÿÿÿÿÿuK
__sig__2o¤o¼oLðt,uûÿÿÿdsÌÿÿÿÿÿÿÿÿ sÎ,ûÿûÿûÿûÿûÿHs@sp4shÏ
	__sig__46H`Ðp<sûÿÿÿÑ
Normalise xüÒ,,ûÿûÿûÿûÿPspDs¼Ó
	__sig__47”¬ÔpLsûÿÿÿüÕ$Convert x to an unnormalised elementÖ,ûÿûÿûÿûÿûÿps`spTsì×
	__sig__48ôØp\sûÿÿÿ¼Ù
Unnormalise@èÚÿÿÿÿÿÿÿÿXsÛ$Convert x to an unnormalised element$¸$Ü,,ûÿûÿûÿûÿxsplst*Ý
	__sig__49,,,Þptsûÿÿÿ¼.ßThe list corresponding to x\1t1à,,ûÿûÿûÿûÿˆsp|sœ8á
	__sig__504:L:âp„sûÿÿÿÌ;äÿÿÿÿÿÿÿÿ€såThe torus term h_r(r)D?æþÿ+,ûÿûÿûÿûÿ sp”sDç
	__sig__51œE´Eèpœsûÿÿÿ\Gé
	TorusTermœI´Iêÿÿÿÿÿÿÿÿ˜sëThe inverse of xN°NøNì,,ûÿûÿûÿûÿŒ‰¸sp¬s¤Rí
	__sig__52ìSTîp´sûÿÿÿVïThe product of x and yXð,,,ûÿûÿûÿûÿœ‰Èsp¼s°\ñ
	__sig__53 _8_òpÄsûÿÿÿè`óTrue iff x and y are equal0bHbô,,ûÿûÿûÿûÿÜˆØspÌsfõ
	__sig__54èfgöpÔsûÿÿÿ8k÷The nth power of x`l˜lø,,ûÿûÿûÿûÿ`tèspÜsäpüpÄrtðtÌuûÿÿÿÔuStandardRepresentationvÿÿÿÿÿÿÿÿÈuw3The degree of the representation with weight lambda¨Àxu;ûÿûÿûÿûÿèuðtÜuzðtäuûÿÿÿ,|ÿÿÿÿÿÿÿÿàu“
ghtector}BReturns the highest weight vector of an irreducible representation\|~ÿÿûÿûÿûÿûÿvðtôuT
	__sig__11`€€ðtüuûÿÿÿHighestWeightVectorÐ ü ‚ÿÿÿÿÿÿÿÿøuƒ0Returns the weight corresponding to the vector v&ˆ&Ð&„ÿÿÿÿûÿûÿûÿlÍvðtv,,…
	__sig__12.,.†ðtvûÿÿÿ1‡XReturns the highest weight of an irreducible representation, with a corresponding vector8„8œ8ˆÿÿÿÿûÿûÿûÿ(vðtv,<‰
	__sig__13==Šðt$vûÿÿÿÜ>‹
HighestWeight@Œÿÿÿÿÿÿÿÿ vRow reduce the matrix g´EÌEäEŽk+,ûÿûÿûÿûÿ@vðt4vtJ
	__sig__140MPMðt<vûÿÿÿøN‘
	RowReducexPP’ÿÿÿÿÿÿÿÿ8vP
__sig__0LonNew”+ûÿûÿûÿûÿXvðtLv”X•
	__sig__15tZœZ–ðtTvûÿÿÿP\—StandardRepresentationNewP_h_˜ÿÿÿÿÿÿÿÿPvv0ûÿûÿûÿûÿàyÈyÔyxz		y
ElementSequence@fzÿÿÿÿÿÿÿÿØy{?generators for Ext (G/G', U) as cocyclic matrices; U is abelianHm`mxm|ûÿûÿûÿøyÈyìysª
definitionsrDuIBThe matrix for the adjoint action of e_alpha (acting on the right)¤
¼
JØûÿûÿûÿûÿ0uðt$uÀM cThe elements representing the fundamental roots for the simple Lie algebra of the given root datum.xNØûÿûÿûÿ@uðt4uO
__sig__3$<Pðt<uûÿÿÿ¼Q
LieRootEltsÔìRÿÿÿÿÿÿÿÿ8uS cThe matrices representing the fundamental roots for the simple Lie algebra of the given root datum.ðTØûÿûÿûÿXuðtLuÀ#U
__sig__4$ø$%VðtTuûÿÿÿx'W
LieRoots(¼()XÿÿÿÿÿÿÿÿPuZýÿØXûÿûÿûÿûÿpuðtdu¼3[
__sig__56D6\6\ðtluûÿÿÿœ8]RootMatrixFunctiont:œ:^ÿÿÿÿÿÿÿÿhuy
	__sig__10tion|=`ýÿØXûÿûÿûÿûÿˆuðt|udAa
__sig__6C0C¸Cbðt„uûÿÿÿ´EÓ
__sig__1`ˆÔ0wLwûÿÿÿœÕAllowableSubgroup¨
À
ÖÿÿÿÿÿÿÿÿHw×.šlengths of orbits and orbit representatives of all k-dimensional subspaces of natural vector space under action of matrix group G defined over prime fieldÜôØjûÿûÿûÿûÿhw0w\wˆÙ
__sig__2ˆ Ú0wdwûÿÿÿ@Û
OrbitsOfSpaces`Üÿÿÿÿÿÿÿÿ`wÝ-/usr/local/Magma2.8/package/GrpPC/pgrps/ops.m		pŒ		Þ
-/usr/local/Magma2.8/package/GrpPC/pgrps/ops.m		pŒ		ßJGiven a p-group G and a list of tuples [<G.i, W.i>] for the generators of G, returns a list of tuples for the pc-generators of G. The tuples can be used to construct a homomorphism G -> W. D indicates the relations which give definitions for the pc-generators. .h.À.àûÿûÿûÿûÿˆwxw|w06á
__sig__07¨7À7âxw„wûÿÿÿ:ãCompleteTupleListX;€;äÿÿÿÿÿÿÿÿ€wå*ŒConstruct the map P->Q of p-groups, given by the list of images of the generators of P. D indicates the definitions of pc generators of P.DÈDæûÿûÿûÿûÿ wxw”wøHèxwœwûÿÿÿKép_homTMêÿÿÿÿÿÿÿÿ˜wë8Construct the inverse homomorphism of h:GrpPC -> GrpPC. RR¨Rìûÿûÿûÿûÿ¸wxw¬w˜Ví
__sig__2XPX˜Xîxw´wûÿÿÿ[ïInvHom\ðÿÿÿÿÿÿÿÿ°wñ0¤Returns the map h^n, the n-th power of the homomorphism h. If the Domain of H is of type GrpPC, n may be an integer, otherwise n must be greater or equal to zero.eäeòûÿûÿûÿûÿÐwxwÄwŒkó
__sig__3ldlœlôxwÌwûÿÿÿÜmõPowHomÀoöÿÿÿÿÿÿÿÿÈwù__sig__4u`uxuu¨ucWeylTorusMatrixFunctionJdÿÿÿÿÿÿÿÿ€ue
´NüNfýÿØXûÿûÿûÿûÿ uðt”u¨Rg
__sig__7SðSThðtœuûÿÿÿViWeylMatrixFunctionðWXjÿÿÿÿÿÿÿÿ˜uk
<\T\lýÿØXûÿûÿûÿûÿ¸uðt¬u,am
__sig__8abbnðt´uûÿÿÿ¬c8return standard presentation H for G and map from G to H`˜ûÿûÿûÿTx(xHx˜
__sig__1x(xPxûÿÿÿÀStandardPresentationtÿÿÿÿÿÿÿÿLxYreturn true and a map from G to H if the p-groups G and H are isomorphic; otherwise false$<ûÿûÿûÿ˜…lx(x`x\(xhxûÿÿÿ¼2/usr/local/Magma2.8/package/GrpPC/pgrps/generate.m¨‘		Ð™		2/usr/local/Magma2.8/package/GrpPC/pgrps/generate.mø‘		Ð™		GeneratepGroupsh"€"˜"M SEAQ 
__sig__1t
Elements*Œreturn s-step definition sets and offsets of dimension n in vector space of dimension m, where we must retain the first Fixed basis elements1D1 ûÿûÿˆxtx|xl8#
DefinitionSets:$ÿÿÿÿÿÿÿÿ€x%jŒConstruct descendants of p-group having p-class at most ClassBound, and order at most p^OrderBound; if Exponent is non-zero, generate descendants of G satisfying this exponent law; if StepSizes is supplied, then construct descendants of order p^(n + s) of a group of order p^n, only for s in StepSizes; by default, all descendants of G are returned; if All = false, only capable ones are returnedNN&ûÿûÿûÿûÿ°x xtx”xàQ'
__sig__1RSS(txœxûÿÿÿ\U)
Descendants|V”V*ÿÿÿÿÿÿÿÿ˜x+j‹Construct descendants of p-group having p-class one larger than G, and order at most p^OrderBound; if Exponent is non-zero, generate descendants of G satisfying this exponent law; if StepSizes is supplied, then construct descendants of order p^(n + s) of a group of order p^n, only for s in StepSizes; by default, all descendants of G are returned; if All = false, only capable ones are returned k¸k,ûÿûÿûÿûÿ¸xtx¬xŒo-
__sig__2püpÄr.tx´xûÿÿÿÌt1
__sig__3u¼uÔuoTorusMatrixFunctionøefpÿÿÿÿÿÿÿÿ°uq6The standard representation of the group of Lie type GÈlr+XûÿûÿûÿûÿÐuðtÄuÄrs
__sig__9t|tœt{

WeylDegreeŒu¤u}
__sig__1\„~Èyôyûÿÿÿ˜
ExtGeneratorsŒ
€ÿÿÿÿÿÿÿÿðyBgenerators for Hom (H_2 (G), U) as cocyclic matrices; U is abelianØð‚ûÿûÿûÿzÈyz,ƒ
__sig__2„ÈyzûÿÿÿT†ÿÿÿÿÿÿÿÿh‡.™return invariants of abelian group Z^2 (G, U) where U is an abelian group; Ext and Hom are generators for Ext (G/G', U) and Hom (H_2 (G), U) respectively`€ˆûÿûÿûÿûÿ(zÈyz,!‰
__sig__4"€"˜"ŠÈy$zûÿÿÿh$“
__sig__6ycles`&‹Ireturn representative cocycles from G to (abelian) U as cocyclic matricesä+ü+Œûÿûÿûÿûÿ8zÈy,zD1
__sig__534Ì4ŽÈy4zûÿÿÿŒ7RepresentativeCocyclesÜ9ÿÿÿÿÿÿÿÿ0z‘=central extensions of (abelian) U by G determined by Cocyclesl>’ÿûÿûÿûÿûÿPzÈyDz¤B”ÈyLzûÿÿÿDE•CentralExtensionslF”F–ÿÿÿÿÿÿÿÿHzŸ
__sig__8rocess°L—9set up process for central extensions of (abelian) U by GÐOèO˜‹ûÿûÿûÿûÿhzÈy\zLS™
__sig__7T”T\UšÈydzûÿÿÿW›CentralExtensionProcess,Yœÿÿÿÿÿÿÿÿ`z-true iff central extension process P is emptyø_ž‹ûÿûÿûÿûÿ¨z€zÈytz¨b Èy|zûÿÿÿe¡8construct next central extension determined by process PgPgˆg¢ûÿ‹ûÿûÿûÿûÿûÿzÈy„zm£
__sig__9mnn¤ÈyŒzûÿÿÿäp¥
NextExtension4t¦ÿÿÿÿÿÿÿÿˆz™,/usr/local/Magma2.8/package/GrpPC/pgrps/spec˜;
__sig__0.8/package/r3/usr/local/Magma2.8/package/GrpPC/central/central.munt-pgrp©-/usr/local/Magma2.8/package/GrpPC/pgrps/aut.m		€§		°-/usr/local/Magma2.8/package/GrpPC/pgrps/aut.mš5/usr/local/Magma2.8/package/GrpPC/pgrps/approximate.m		Ø°		›5/usr/local/Magma2.8/package/GrpPC/pgrps/approximate.m		Ø°		œDõestimate size of orbit of U under G by constructing at most MaxSize elements of orbit and counting at most NumberCoincidences coincidences; return lower bound, upper bound, and estimate of size; if no estimate can be made, the function returns 0ètjûÿûÿ|vlvpv$Ÿlvxvûÿÿÿp% 
EstimateOrbit(¡ÿÿÿÿÿÿÿÿtv¢^[U is subspace; A is matrix group describing action of G on U; assume 1-1 correspondence between generators of G and A; find elements of G whose action fixes U and estimate size of orbit of U under A; construct at most MaxSize orbit elements or until we find NumberCoincidences repetitions; if OrderCheck is true, report the order of the stabiliserT=|=£tjjjj”vlvˆvdA¤
__sig__1C0C¸C¥lvvûÿÿÿ´E$|H|ûÿÿÿØ[The trace of the r-th power Frobenius endomorphism, where E is defined over a finite field.¨&ûÿûÿûÿûÿt|\|$|P|”
__sig__2$|X|ûÿÿÿd7The trace of Frobenius of the base extension of E to K.Ph€q&ûÿûÿûÿûÿ°Ùl|$|`|€
__sig__3 À$|h|ûÿÿÿ7The trace of Frobenius of the base extension of E to K.Ld„q&ûÿûÿûÿûÿ||$|p|0!
__sig__4"„"œ"$|x|ûÿÿÿl$ &}The order of the group of rational points on the degree r extension of the base ring, where E is defined over a finite field.0,!&ûÿûÿûÿûÿ”|Œ|$|€|x1"
__sig__54˜5ð5#$|ˆ|ûÿÿÿÀ7$&}The order of the group of rational points on the degree r extension of the ring of E, where E is defined over a finite field.ð<%fûÿûÿûÿûÿ¤|œ|$||8@&
__sig__6BhB¨B'$|˜|ûÿÿÿHE("lThe order the group of rational points on E over K, where E is defined over the rationals or a finite field.JK)q&ûÿûÿûÿûÿ…¬|$| |ìO*
__sig__7QLQdQ+$|¨|ûÿÿÿðR, cThe Euler factor of E, i.e. the reciprocal characteristic polynomial of the Frobenius endomorphism.ðWX-&ûÿûÿûÿûÿÌ|¼|$|°|´\.
__sig__8^$_<_/$|¸|ûÿÿÿì`0
EulerFactorÔaìa1ÿÿÿÿÿÿÿÿ´|21The Euler factor of the base extension of E to K.Œf¤f3q&ûÿûÿûÿûÿÄ„Ô|$|È|Ll4
__sig__9mdm|m5$|Ð|ûÿÿÿÀo6The zeta function of E.hss¨s8
	__sig__10 uHu9$|à|ûÿÿÿ„vœvÜv¦ApproximateStabiliserøH§ÿÿÿÿÿÿÿÿŒv¨4/usr/local/Magma2.8/package/GrpPC/pgrps/attributes.m·		°»		Ê0/usr/local/Magma2.8/package/GrpPC/pgrps/finger.m¸		ø{	ä»ç¬
stepSizeShSS¯fixed­multiplicatorRankÐVW®
nuclearRank¸XÐXç
__sig__1roup2[²$qCompute the automorphism group of G; CharacteristicSubgroups is a sequence of known characteristic subgroups of Ga,a³FûÿûÿûÿûÿÔvÀvÈväd´
__sig__0eÌeäeµÀvÐvûÿÿÿg¶AutomorphismGroupPGroup<k·ÿÿÿÿÿÿÿÿÌv¸[Compute the automorphism group of G; C is a sequence of known characteristic subgroups of Gs¨sº
__sig__1u uHu»Àvèvûÿÿÿ„vÉ2«Returns the sequence of integral points on the quartic curve y^2 = ax^4 + bx^3 + cx^2 + dx + e, where Q = [a,b,c,d,e] and P is a given rational point not equal to [0,1,0].´ÔÍIntegralQuarticPointsôÎÿÿÿÿÿÿÿÿ(Ï*ˆReturns the sequence of integral points on the quartic curve y^2 = ax^4 + bx^3 + cx^2 + dx + e, where Q = [a,b,c,d,e] and e is a square.8XÐÿûÿûÿûÿûÿH¸~<€Ñ
__sig__6x˜Ò¸~DûÿÿÿðÓ2«Returns the sequence of S-integral points on the quartic curve y^2 = ax^4 + bx^3 + cx^2 + dx + e, where Q = [a,b,c,d,e] with square a, and where S is a sequence of primes.„"œ"ÔÿÿûÿûÿûÿûÿX¸~L'Õ
__sig__7((¨(Ö¸~Tûÿÿÿ,×SIntegralQuarticPoints0.ØÿÿÿÿÿÿÿÿPÙ*ŽReturns the sequence of S-integral points on the Ljunggren curve a*y^2 = b*x^4 + cx^2 + d, where Q = [a,b,c,d], and S is a sequence of primes.;Úÿÿûÿûÿûÿûÿp¸~d >Û
__sig__8?`?x?Ü¸~lûÿÿÿhAÝSIntegralLjunggrenPointsDPDÈDÞÿÿÿÿÿÿÿÿhß,’Returns the sequence of S-integral points to the Desboves curve a*y^3 + b*x^3 + c*x*y + d = 0, where C = [a,b,c,d], and S is a sequence of primes.4O\Oàÿÿûÿûÿûÿûÿˆ¸~|ðRá
__sig__9T(T@Tâ¸~„ûÿÿÿ`VãSIntegralDesbovesPoints˜Xäÿÿÿÿÿÿÿÿ€å2/usr/local/Magma2.8/package/CrvEll/supersingular.mÄ		Ç		æ2/usr/local/Magma2.8/package/CrvEll/supersingular.mXÄ		Ç		ç5A representative supersingular elliptic curve over K.eèq&ûÿûÿûÿûÿ¨˜œügé
__sig__0k¤k¼kê˜¤ûÿÿÿmëSupersingularEllipticCurveènoìÿÿÿÿÿÿÿÿ í1Returns true if ordinary, false if supersingular.0vHvî&ûÿûÿûÿûÿÀ˜´èy§-/usr/local/Magma2.8/package/Process/process.m		ØÑ		¨-/usr/local/Magma2.8/package/Process/process.m		ØÑ		©(True if no objects remain in the processdªûÿûÿûÿûÿ°z z¤z`«
__sig__08P¬ z¬zûÿÿÿˆ­)Advance to the next object in the process¨À®ûÿûÿûÿûÿûÿÀz z´zÀ¯
__sig__1¸à° z¼zûÿÿÿ²ÿÿÿÿÿÿÿÿ¸z³%Get the current object of the processL#´ÿÿûÿûÿûÿûÿØz zÌz|'µ
__sig__2(À()¶ zÔzûÿÿÿ0,¸ÿÿÿÿÿÿÿÿÐz¹.The Label of the current object of the processÐ4ºÿÿûÿûÿûÿûÿðz zäz:»
__sig__3;;0;¼ zìzûÿÿÿ<½
CurrentLabel=Ð=¾ÿÿÿÿÿÿÿÿèzi$QAn affine plane constructed by the technique of derivation w.r.t. a Baer subplane		ÿÿÿÿˆEj$ÂÁÀûÿûÿ°‘ ‘¤‘o$>A translation plane from PG(2, q) by derivation w.r.t. an oval\Np$ÂÁÀûÿûÿÈ‘ ‘¼‘s.m¿./usr/local/Magma2.8/package/CrvEll/CrvEll.spec€TÄ__sig__0.8/package/CrvEllK __sig__0.8/package/CrvEllh 1/usr/local/Magma2.8/package/SchCrvHyp/CrvHyp.spec/tÐÎ		0Ï		 
__sig__0ge/CrvEll/sÀ0/usr/local/Magma2.8/package/CrvEll/base_change.mbhÏ		ØÑ		Á0/usr/local/Magma2.8/package/CrvEll/base_change.me¸Ï		ØÑ		Â2Base extension of the isogeny m to the base ring K´gÌgÃýÿûÿûÿûÿûÿ{{{LmÅ{{ûÿÿÿoÆ
LiftIsogenyqÈrÇÿÿÿÿÿÿÿÿ{‘PeriodseÈ6Base extension of the isomorphism m to the base ring K¨xQ4y@yûÿÿÿ	Sÿÿÿÿÿÿÿÿ<ybClassTwoAbelianPGrouproupT&xOrder of automorphism group of abelian p-group G where a = [a[1], a[2], a[3], ...] and G = C_a[1] x C_a[2] x C_a[3] ... ÜôUûÿûÿûÿûÿ\y4yPyˆV
__sig__1ˆ W4yXyûÿÿÿ@X#OrderAutomorphismGroupAbelianPGroupøYÿÿÿÿÿÿÿÿTyZ*‹Return the number of subgroups of each non-trivial order in the abelian p-group G where a = [a[1], a[2], ...] and G = C_a[1] x C_a[2] x ...„"œ"[ûÿûÿûÿûÿty4yhy'\
__sig__2((¨(]4ypyûÿÿÿ,^NumberOfSubgroupsAbelianPGroupÀ._ÿÿÿÿÿÿÿÿly`5/usr/local/Magma2.8/package/GrpPC/count-pgrps/count.m		Ü		a5/usr/local/Magma2.8/package/GrpPC/count-pgrps/count.m		Ü		c&xCount all p-class 2 d-generator groups of order p^(d + s); if Exponent is true, count those groups which have exponent p??¨?dûÿûÿûÿûÿ¨y˜y„yŒyEe
__sig__0EFFf„y”yûÿÿÿ¨Hg
ClassTwoIJJhÿÿÿÿÿÿÿÿyi(†Count all p-class 2 d-generator groups of order p^(d + s) for s in Step; if Exponent is true, count those groups which have exponent pPSjûÿûÿûÿûÿ¸y°y„y¤y¨Wk
__sig__1YHYYl„y¬yûÿÿÿÐ[m eCount all p-class 2 d-generator groups; if Exponent is true, count those groups which have exponent p\anûÿûÿûÿûÿÀy„y´yeo
__sig__2eüefp„y¼yûÿÿÿŒgU!
__sig__0Y!
__sig__1L ,ûÿÿÿu _return indexed set of elements of G listed in the order used by ExtGenerators and HomGeneratorsHu`uxuxÈyÜyûÿÿÿÜv·CurrentØw÷MGiven a Map as composition of homs, construct the composition map explicitly.œøûÿûÿûÿûÿèwxwÜwÜúxwäwûÿÿÿû
BuildHom4Lüÿÿÿÿÿÿÿÿàwý2/usr/local/Magma2.8/package/GrpPC/pgrps/parallel.møÝ		Ðå		NAn indexed set containing the points at infinity lying on the elliptic curve EH€0ûÿûÿûÿûÿ~~ø}ü}ƒXAn indexed set containing the points at infinity in the point set H of an elliptic curve\t„f0ûÿûÿûÿûÿ~ø}~…
__sig__1ô†ø}~ûÿÿÿl‡DóReturns the isogeny E->F defined by the polynomials psi, phi and omega, taking (x, y) to (phi / psi^2, omega / psi^3). Psi and phi must be univariate (although elements of a multivariate polynomial ring) and the polynomials need not be coprimeðˆKKK&&ûÿûÿûÿûÿ(~ø}~À#‰
__sig__2$ø$%Šø}$~ûÿÿÿx'‹,/usr/local/Magma2.8/package/CrvEll/periods.mâ		`é		Œ,/usr/local/Magma2.8/package/CrvEll/periods.mâ		`é		8The real and the complex period of the elliptic curve E.4”5ì5Ž&ûÿûÿûÿûÿ´Ð@~0~4~L:
__sig__0;T;|;0~<~ûÿÿÿ¼<’ÿÿÿÿÿÿÿÿ8~”&VûÿûÿûÿûÿX~0~L~¸C•
__sig__1E\E„E–0~T~ûÿÿÿ$G—

RealPeriod,ItI˜ÿÿÿÿÿÿÿÿP~™&The elliptic logarithm of the point P.XOšeVûÿûÿûÿûÿp~0~d~ìR›
__sig__2T$T<Tœ0~l~ûÿÿÿ\VEllipticLogarithmLX”Xžÿÿÿÿÿÿÿÿh~©
__sig__0rithm ]Ÿ&zGiven a point P on an elliptic curve, which must be a minimal model, and a prime p, returns the p-adic elliptic logarithm.b¨b e’ûÿûÿûÿûÿˆ~0~|~pf¡
__sig__3g°gÈg¢0~„~ûÿÿÿ¸k£pAdicEllipticLogarithmm¤ÿÿÿÿÿÿÿÿ€~¥3/usr/local/Magma2.8/package/CrvEll/reduction_type.mè		`é		¦3/usr/local/Magma2.8/package/CrvEll/reduction_type.màè		`é		ª˜~¤~ûÿÿÿ¤x/h€Generate tree of d-generator p-class <= c p-groups of order at most p^OrderBound; if Exponent is non-zero, generate groups satisfying this exponent law; if StepSizes is supplied, then construct extensions of order p^(n + s) of a group of order p^n only for s in StepSizes; by default, all groups satisfying these properties are returned; if All = false, only capable ones are returnedd0ûÿûÿûÿûÿÈxtx¼x2txÄxûÿÿÿ3
GeneratepGroups4ÿÿÿÿÿÿÿÿÀx52/usr/local/Magma2.8/package/GrpPC/count-pgrps/specdq3/usr/local/Magma2.8/package/GrpPC/central/central.m69/usr/local/Magma2.8/package/GrpPC/count-pgrps/backtrack.m ì		øó		75/usr/local/Magma2.8/package/GrpPC/count-pgrps/fixed.m		øó		85/usr/local/Magma2.8/package/GrpPC/count-pgrps/fixed.m		øó		94Return number of subspaces of dimension s fixed by gM:kûÿûÿûÿûÿyðxàxäx<àxìxûÿÿÿd=NumberOfFixedSpacesR>ÿÿÿÿÿÿÿÿèx?4Return number of subspaces of dimension s fixed by g@ûÿûÿûÿûÿyàxüxœXA
__sig__1dBàxyûÿÿÿC7/usr/local/Magma2.8/package/GrpPC/count-pgrps/classes.møó		D7/usr/local/Magma2.8/package/GrpPC/count-pgrps/classes.møó		EHReturn representatives of conjugacy classes of GL (n, q) and their sizesdç)I;Fûÿûÿûÿûÿ yyyG
__sig__0{‹W;HyyûÿÿÿdIConjugacyClassesGL„tJÿÿÿÿÿÿÿÿyK4/usr/local/Magma2.8/package/GrpPC/count-pgrps/auto.mò		øó		L9/usr/local/Magma2.8/package/GrpPC/count-pgrps/subgroups.m		ˆò		øó		M9/usr/local/Magma2.8/package/GrpPC/count-pgrps/subgroups.màò		øó		NReturn order of GL (n, p)dvs£:OûÿûÿûÿûÿDy4y8yROrderGLþ4/usr/local/Magma2.8/package/GrpPC/pgrps/parameters.mó		û		ÿ0/usr/local/Magma2.8/package/GrpPC/pgrps/shadow.ms.mô		û		s
Cocycle2.8/package/±Advances8/usr/local/Magma2.8/package/GrpPC/pgrps/stab-of-spaces.màô		û		8/usr/local/Magma2.8/package/GrpPC/pgrps/stab-of-spaces.m8õ		û		w
__sig__0!__sig__0,“Subgroup of GL(d, F) which fixes sequence of subspaces of natural vector space; also return generators for largest unipotent subgroup of stabiliserdjûÿûÿûÿxxxxxûÿÿÿStabiliserOfSpacesdÔu ;ÿÿÿÿÿÿÿÿx	2/usr/local/Magma2.8/package/GrpPC/pgrps/standard.m¸÷		û		
2/usr/local/Magma2.8/package/GrpPC/pgrps/standard.mø		û		æ0ModulemJP is p-covering group of H; M is relative multiplicator; K is class c quotient; H is class c - 1 quotient; if Relative is non-negative, then it is the relative step size; otherwise step size is log of #K div #H; return matrix of (relative) allowable subgroup from P to Kwûÿûÿûÿûÿ<x(x0xøv
__sig__0dv(x8xûÿÿÿ_&zThe height pairing lattice of the free quotient of the Mordell-Weil group of E, together with a map to the elliptic curve.$`&Ôûÿûÿûÿˆ}L}|} bL}„}ûÿÿÿ°cMordellWeilLattice0dÿÿÿÿÿÿÿÿ€}q
__sig__5ice¬e&zThe height pairing lattice of the free quotient of the Mordell-Weil group of E, together with a map to the elliptic curve.$f&Ôûÿûÿûÿ°} }L}”}g
__sig__3lŒhL}œ}ûÿÿÿèiHeightPairingLattice!è!jÿÿÿÿÿÿÿÿ˜}k[The height pairing lattice of a sequence of independent points on an elliptic curve over Q.\*t*lÿeûÿûÿûÿ¸}L}¬}ü/m
__sig__41\1t1nL}´}ûÿÿÿ,6o&yThe regulator of the sequence S of points on an elliptic curve over Q, i.e. the determinant of the height pairing matrix.ä;ü;pÿeVûÿûÿûÿûÿÐŒÈ}L}¼}D?rL}Ä}ûÿÿÿ,A
__sig__0dentCsBêReturns true if and only if P is linearly independent of Q over the integers; if false then the second return value is a vector giving a linear combination of the points which is torsion (and thus in the kernel of the height pairing).0OXOtee9ûÿûÿûÿè}Ø}L}Ì}ìRu
__sig__6T$T<TvL}Ô}ûÿÿÿ\VwIsLinearlyIndependent”XxÿÿÿÿÿÿÿÿÐ}yFúReturns true if and only if the sequence of points is linearly independent over the integers; if false then the second return value is a vector giving a linear combination of the points which is torsion (and thus in the kernel of the height pairing).øefzÿe9ûÿûÿûÿh€ð}L}ä}¸k{
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FunctionFieldHm 
Discriminantdn 
	BadPrimesH`!ûÿûÿûÿûÿ´‹„˜ƒ„$!˜ƒ„ûÿÿÿ<!ûÿûÿûÿûÿ|Š „˜ƒ„p	!
	__sig__11!˜ƒ$„ûÿÿÿ!DTrue iff pt is coercible into Jacobian J; if true, also return J!pt Ð
!.Extends the base field of the Jacobian J to F.ô!ýÿûÿûÿûÿûÿÜƒ4„˜ƒ(„À!
__sig__8Ô!˜ƒ0„ûÿÿÿì!ûÿûÿûÿ¼ŠD„˜ƒ8„¼	!
	__sig__12Ð	!˜ƒ@„ûÿÿÿè	!DòReturns all rational points on the Jacobian J over a finite field, or all points on a Jacobian of a genus 2 curve of the form y^2 = f(x), defined over the rationals, up to a bound on the naive Kummer surface height set by the parameter Bound.è!0ûÿûÿûÿûÿl„T„˜ƒH„Ô$!
	__sig__13H5!˜ƒP„ûÿÿÿp5!DòReturns all rational points on the Jacobian J over a finite field, or all points on a Jacobian of a genus 2 curve of the form y^2 = f(x), defined over the rationals, up to a bound on the naive Kummer surface height set by the parameter Bound.R!0ûÿûÿûÿûÿ|„d„˜ƒX„”^!
	__sig__14¨^!˜ƒ`„ûÿÿÿÀ^!$sFind all points on J with first component a and degree d. So far only for genus 2 and curve of the form y^2 = f(x)._!0ûÿûÿûÿûÿÌŠt„˜ƒh„°v!
	__sig__15¬!˜ƒp„ûÿÿÿ¬ƒLmo ;The hyperelliptic curve given by the polynomial sequence s.qÈrp ÿgûÿûÿûÿûÿÐÈ¨¼Hvq 
__sig__0ww¨wr ¨ÄûÿÿÿLys /The hyperelliptic curve given by y^2 + h y = f.¬{<|Ä|u 
__sig__1~~°~v ¨Ôûÿÿÿ€°€È€Ï 
	__sig__22\„Ð ¨<ƒûÿÿÿ˜Ñ  dReturns the indexed set of all rational points on the hyperelliptic curve C with given x-coordinate.0Ò +g0ûÿûÿûÿûÿL„Pƒ¨DƒDÓ 
	__sig__23H`Ô ¨LƒûÿÿÿŒÕ  dReturns the indexed set of all rational points on the hyperelliptic curve C with given x-coordinate.ìÖ +g0ûÿûÿûÿûÿ\„`ƒ¨Tƒ× 
	__sig__24Ø ¨\ƒûÿÿÿ€Ù WThe zeta function of C, with finite base field, as a rational function of one variable.˜!°!è!Ú gSûÿûÿûÿûÿxƒpƒ¨dƒp%Û 
	__sig__25è'(Ü ¨lƒûÿÿÿ¼*Ý ZThe zeta function of the base extension of C to K, as a rational function of one variable.Ü01Þ qgSûÿûÿûÿûÿ€ƒ¨tƒ¼7ß 
	__sig__26Ä9Ü9à ¨|ƒûÿÿÿ,;á :ÊFind a random point on C. Base field must be finite. If the set of all points on C has already been computed, this gives a truly random point, otherwise the ramification points have a slight advantage. àCDâ ghûÿûÿûÿûÿ”×ƒ¨„ƒ¤Hã 
	__sig__27üIJä ¨ŒƒûÿÿÿPMå 0/usr/local/Magma2.8/package/SchCrvHyp/jacobian.mOÐ
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TorsionMap$é 

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TorsionGroupYŒYí HeightConst$ì 
TorsionBound]p]ø __sig__9ÿÿÿ!JExtends the finite base field of the Jacobian J to its degree n extension.	¼ç!
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	 Ï		ä»çú "The base field of the Jacobian J. ˆf fû ýÿûÿûÿûÿûÿÄ‹ôƒ˜ƒèƒHlü 
__sig__3m`mxmý ˜ƒðƒûÿÿÿ¼oþ "The base field of the Jacobian J. üs4tÿ ýÿûÿûÿûÿûÿÔ‹„˜ƒøƒlw!
__sig__4xŒx¤x!˜ƒ„ûÿÿÿTz÷ ûÿûÿûÿûÿ„äƒ˜ƒØƒ4ù ˜ƒàƒûÿÿÿÈ
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	§,Returns a string "Good", "Additive", "Split multiplicative" or "Nonsplit multiplicative" describing the reduction type of a minimal model for E.lA¨&ûÿûÿûÿûÿ¨~˜~œ~«
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	¯QTrue if and only if the denominator of n has support in the sequence of primes S.d­W3°ÿûÿûÿûÿûÿØ~È~¸~¼~±
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IsSIntegral´ÿÿÿÿÿÿÿÿÀ~µ.True if and only if S is a sequence of primes.;¶ÿûÿûÿûÿûÿè~à~¸~Ô~d·
__sig__1VP9¸¸~Ü~ûÿÿÿE0S9¹ITrue if and only if P is not the identity and has S-integral coordinates.ºeûÿûÿûÿûÿð~¸~ä~ Z»
__sig__2d¼¸~ì~ûÿÿÿ½:ÈGiven an elliptic curve over the rationals, returns the sequence of integral points, modulo [-1], followed by a sequence of tuple sequences representating the points as a linear combination of points.d¾&ûÿûÿûÿ¸~ô~¿
__sig__3dóŽW;À¸~ü~ûÿÿÿÁ
IntegralPoints:Âÿÿÿÿÿÿÿÿø~Ã@åGiven an elliptic curve over the rationals and a sequence S of primes, returns the sequence of S-integral points, modulo [-1], followed by a sequence of tuple sequences representating the points as a linear combination of points.Äÿ&ûÿûÿûÿ¸~Å
__sig__4xÆ¸~ûÿÿÿyÇ
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__sig__1pticCurveÊÿÿûÿûÿûÿûÿ@0¸~$Ë
__sig__5|Ì¸~,ûÿÿÿ7&Sûÿûÿûÿûÿì|ä|$|Ø|:2The zeta function of the base extension of E to K.5ž©9;q&Sûÿûÿûÿûÿhƒô|$|è|d<
	__sig__11;=$|ð|ûÿÿÿ3£Ñ8>./usr/local/Magma2.8/package/CrvEll/functions.m	°3
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	@5The affine equation order in the function field of E.×¨í:A&âûÿûÿûÿûÿ ‚}ü|}B
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X;Cü|}ûÿÿÿdD0£The bivariate polynomial ring associated with E, i.e. the coordinate ring of the affine space of E. The map to the equation order is returned as a second argument.E&Lûÿûÿûÿ}ü|}RF
__sig__1dGü|}ûÿÿÿH$The defining affine polynomial of E.dI&Kûÿûÿûÿûÿ¤‹,}ü| }J
__sig__2dfýù7Kü|(}ûÿÿÿLDefiningEquation ZMÿÿÿÿÿÿÿÿ$}N$The defining affine polynomial of E.*œ6;O&KûÿûÿûÿûÿlD}ü|8}P
__sig__3ËW;Qü|@}ûÿÿÿdR,/usr/local/Magma2.8/package/CrvEll/heights.m/
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HeightPairingìtU^The height pairing matrix of the sequence S of points on an elliptic curve over the rationals.Vÿeûÿûÿûÿûÿp}`}L}T}$wW
__sig__0dvXL}\}ûÿÿÿYHeightPairingMatrixZÿÿÿÿÿÿÿÿX}[ _The height pairing matrix of the Mordell-Weil group of the elliptic curve E over the rationals.Ôxðo:\&ûÿûÿûÿûÿÀŒx}L}l}Ôx]
__sig__1S;^L}t}ûÿÿÿÀü];a
__sig__21 TwistsupÉýÿûÿûÿûÿûÿ,{{ {lAÊ
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	â__sig__4cientsivariateÐ"gReturns the multivariate polynomial whose roots determine the points of order 2 on the elliptic curve EdîÕ4Ñ&KûÿûÿûÿûÿL{<{@{Ò
__sig__08îÕ4Ó<{H{ûÿÿÿdÔ TwoTorsionPolynomialMultivariate;îÕ4ÕÿÿÿÿÿÿÿÿD{×*ŒReturns the two-element sequence defining a simplified Weierstrass model of the elliptic curve. Note that 2 and 3 must not be zero divisors.Ù
__sig__3{X{dØ<{d{ûÿÿÿÚLReturns an elliptic curve over K with coefficients coerced to K by the map jÛõÿ&&ûÿûÿûÿûÿ,„t{<{h{°v ;Ü
__sig__2DZÝ<{p{ûÿÿÿÖ&ûÿûÿûÿûÿ`{<{\{dÞWeierstrassCoefficientsßÿÿÿÿÿÿÿÿX{àOThe set of rational points of the elliptic curve E with base ring extended to Kdáýÿ&ûÿûÿûÿûÿ(ƒŒ{<{€{ã<{ˆ{ûÿÿÿä,/usr/local/Magma2.8/package/CrvEll/cremona.m;
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	æ%The Cremona database reference for E.ç&5ûÿûÿûÿûÿ´{¤{”{˜{$xè
__sig__0dvé”{ {ûÿÿÿêCremonaReferenceðo:ëÿÿÿÿÿÿÿÿœ{ì%The Cremona database reference for E.dî
__sig__1> é&ûÿûÿûÿûÿÐ€ô€HA *ˆThe elliptic curve E specified by the genus one curve C and the point P, followed by the birational map C -> E, and the morphism E -> C.xB íé&ûÿûÿ8ËÐ€ÀC 
__sig__3¤¼D Ð€ûÿÿÿÜ>/home/was/magma/local/init.m?
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HeightConstantS!(The quadratic twist of the curve C by d.I >The order of E using the methods of Schoof, Elkies, and Atkin.D(J &ûÿûÿûÿûÿ@0$ü-N ÿÿÿÿÿÿÿÿ(O >The order of E using the methods of Schoof, Elkies, and Atkin.Œ7P fûÿûÿûÿûÿH<|;R Dûÿÿÿ¼<$./usr/local/Magma2.8/package/CrvMod/CrvMod.specD?S 4/usr/local/Magma2.8/package/CrvEll/SEA/trace_table.mC
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	T 7/usr/local/Magma2.8/package/CrvEll/SEA/torsion_module.m I
	U //usr/local/Magma2.8/package/CrvEll/SEA/schoof.modulV //usr/local/Magma2.8/package/CrvEll/SEA/elkies.mˆMW ./usr/local/Magma2.8/package/CrvEll/SEA/atkin.mXPX 0/usr/local/Magma2.8/package/CrvEll/SEA/lercier.mRE
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	Y 0/usr/local/Magma2.8/package/CrvEll/SEA/menezes.mU`E
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	l 	] Return f(g) mod hÐ`è`^ ûÿûÿûÿûÿ€lt°d_ 
__sig__0e˜e°e` l|ûÿÿÿÐfa ModularCompositioni˜jb ÿÿÿÿÿÿÿÿxc <ÔReturn the r-th iterate of composition of f with itself modulo h, computed using the Brent-Kung machinery. Assumes that f defines an endomorphism of the quotient by h, so that modular composition is well-defined.wlwd ûÿûÿûÿûÿ˜lŒÜze 
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__sig__0dent¤ JTrue if and only if P and Q form a basis of the $n$-torsion subgroup of E.x eeûÿûÿûÿûÿx€p€@€d€À 
__sig__2¤¼ @€l€ûÿÿÿÜ WTrue if and only if S is a sequence of n-torsion points linearly independent over Z/nZ.<\| ÿeûÿûÿûÿûÿ€€@€t€T 
__sig__3`€  @€|€ûÿÿÿ! +/usr/local/Magma2.8/package/CrvEll/twists.m€L
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	# The quadratic twist of E by d.'$ þÿ&&ûÿûÿûÿûÿP…˜€ˆ€Œ€¬,& ˆ€”€ûÿÿÿl/' .The quadratic twists of E over a finite field.ô3( &&ûÿûÿûÿûÿp…¨€ˆ€œ€Ü9) 
__sig__1:ä:ü:* ˆ€¤€ûÿÿÿ\<+ 
QuadraticTwists|=, ÿÿÿÿÿÿÿÿ €- $The twists of E over a finite field.B¤B. &ûÿûÿûÿûÿÀ€ˆ€´€$G/ 
__sig__2H,ItI0 ˆ€¼€ûÿÿÿðK2 ÿÿÿÿÿÿÿÿ¸€3 ./usr/local/Magma2.8/package/CrvEll/transform.m	8U
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	H SEAee\U5 .˜If C is a genus 1 curve of odd degree returns true, and if so, also returns the corresponding elliptic curve E, the isomorphism C -> E, and its inverse.\ø\ ]6 g&ûÿà€Ð€Ô€ˆa7 
__sig__0b`bxb8 Ð€Ü€ûÿÿÿàd9 (…The elliptic curve E specified by the genus one hyperelliptic curve C of odd degree, followed by the morphism C -> E and its inverse.èk: g&ûÿûÿø€ð€Ð€ä€¼o; 
__sig__1rÜrs< Ð€ì€ûÿÿÿu= 0¡The elliptic curve E specified by the genus one curve C, assuming that the equation of C is in elliptic Weierstrass form, followed by the maps C -> E and E -> C.ôz{? 
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IsOrdinaryŒ¤òÿÿÿÿÿÿÿÿ¸ó/Returns true if supersingular, false otherwise.°üô&ûÿûÿûÿûÿØ˜Ì¼õ
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IsSupersingularLøÿÿÿÿÿÿÿÿÐ	 
__sig__1hemerDù6Returns false if shown to be ordinary, true otherwise.€ú&ûÿûÿûÿûÿð˜ä,!û
__sig__3"€"˜"ü˜ìûÿÿÿh$ýIsProbablySupersingularÐ&þÿÿÿÿÿÿÿÿèE 
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	 ZReturns the scheme corresponding to the subgroup of G containing the n-torsion points of G<,< ûÿûÿûÿûÿ€€€t? €€ûÿÿÿdA TorsionSubgroupSchemeD ÿÿÿÿÿÿÿÿ€ $pGiven a subgroup G of an elliptic curve E, returns an elliptic curve E1 and an isogeny I : E -> E1 with kernel GM MÀM &ûÿûÿûÿ(€€€ Q
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__sig__0Factored .™Given a subgroup G of an elliptic curve E, returns an elliptic curve E1 and a sequence of isogenies whose product is an isogeny I : E -> E1 with kernel Gˆ\°\ &ûÿûÿûÿ8€€,€Xa 
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	 "hThe Sylow p-subgroup of the group of rational points on E over a finite field, followed by the map to E.op$p &“ûÿûÿûÿP€@€D€¤u @€L€ûÿÿÿw %The rational 2-torsion subgroup of E.Hy “ûÿûÿûÿì‹`€@€T€D~"*ŠThe transformation of C given by the scaling factor e and shifting polynomial u, returning the codomain curve followed by the isomorphism.D"þÿgg–ûÿûÿûÿtˆlˆÄ‡`ˆx"
__sig__6Œ"Ä‡hˆûÿÿÿ¤",“The transformation on curves of genus g given by the shift polynomial u, of the abscissa, returning the codomain curve followed by the isomorphism."gg–ûÿûÿûÿ„ˆ|ˆÄ‡pˆÈ"
__sig__7Ü"Ä‡xˆûÿÿÿô "$pThe transformation of C given by the scaling factor e, returning the codomain curve followed by the isomorphism.	Ø	!"þÿgg–ûÿûÿûÿŒˆÄ‡€ˆ
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__sig__9!€!'"Ä‡˜ˆûÿÿÿØ$("HackobjCoerceHomCrvHyp`5)"ÿÿÿÿÿÿÿÿ”ˆ*"
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	__sig__10¬K-"Ä‡°ˆûÿÿÿR."HackobjPrintHomCrvHypäY/"ÿÿÿÿÿÿÿÿ¬ˆ0"
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	__sig__11è^3"Ä‡Èˆûÿÿÿ_4"HackobjPrintMapCrvHyp(g5"ÿÿÿÿÿÿÿÿÄˆ7"””ûÿûÿûÿûÿìˆäˆÄ‡Øˆ°ƒ8"
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GammaDivPol$î -The dimension of the jacobian J as a variety.|cï ûÿûÿûÿûÿ¼ÅÄƒ˜ƒ¸ƒÔfð 
__sig__0gh,hñ ˜ƒÀƒûÿÿÿlò :ÌReturns the sequence of primes where the given model of the curve of J has bad reduction. J must be defined over a number field, and the defining polynomials of its curve must have integral coefficients. vHvó ûÿûÿûÿûÿÔƒ˜ƒÈƒèyô 
__sig__1zøz{õ ˜ƒÐƒûÿÿÿH~ö 7Extends the base field of the Jacobian J by applying j.`l
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__sig__8©1” ¨L‚ûÿÿÿÜõg:•  dThe hyperelliptic curve C corresponding to the elliptic curve E, followed by the isomorphism E -> C.?– &gûÿûÿûÿ`‚¨T‚˜ ¨\‚ûÿÿÿ2;¥ 
	__sig__12veOfGenus™ .—Returns true if and only if the polynomial sequence of length two defines a hyperelliptic curve. In this case, the curve is returned as a second value.dâõ:š ÿgûÿûÿûÿp‚¨d‚› 
	__sig__10Â 3œ ¨l‚ûÿÿÿd 0¢Returns true if and only if the polynomial sequence of length two defines a hyperelliptic curve of genus g. In this case, the curve is returned as a second value.ž ÿgûÿûÿûÿ€‚¨t‚ÈMŸ 
	__sig__11d  ¨|‚ûÿÿÿ¡ IsHyperellipticCurveOfGenus”S¢ ÿÿÿÿÿÿÿÿx‚£ 0The function field of the hyperelliptic curve C.W¤ gàûÿûÿûÿûÿ˜‚¨Œ‚àY¦ ¨”‚ûÿÿÿ§ 'Returns the affine equation order of C.(Z¨ gâûÿûÿûÿûÿ¨‚¨œ‚LZ© 
	__sig__13dª ¨¤‚ûÿÿÿ« 4Evaluates the homogeneous equation of C at (a,b,c). h^¬ þÿþÿþÿgþÿûÿûÿûÿûÿÈ‚¸‚¨¬‚­ 
	__sig__14„t® ¨´‚ûÿÿÿK!ûÿûÿûÿûÿ4…˜ƒ(…LM!˜ƒ0…ûÿÿÿ(N!
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__sig__2T!þÿggûÿûÿûÿûÿ`…X…D…L…ìV!D…T…ûÿÿÿÜW!?The quadratic twist of the curve C defined over a finite field.œ"´"L#X!ggûÿûÿûÿûÿh…D…\…|'Z!D…d…ûÿÿÿx*[!@The quadratic twists of the curve C defined over a finite field../p/\!gûÿûÿûÿûÿx…D…l…`6]!
__sig__278p8^!D…t…ûÿÿÿP:_!(True if and only if C2 is a quadratic twist of C1 over their common base field. If so, the second value is the twisting factor.à>0?H?`!ggþÿûÿûÿûÿˆ…D…|… Da!
__sig__3E E¸Eb!D…„…ûÿÿÿ`Gc!IsQuadraticTwistIÐIèId!ÿÿÿÿÿÿÿÿ€…e!4¯True if and only if C1 and C2 are isomorphic over their common base field, followed by a transformation giving the isomorhism as a second value when the curves are isomorphic.€T˜T`Uf!gg–ûÿûÿûÿŒï …D…”…Yg!
__sig__4[€[ [h!D…œ…ûÿÿÿt]i!2§The automorphism group G of C as a permutation group. The second value is a map from G to the group of automorphisms of C. The third value gives the action C x G -> C.e,eDej!gæûÿûÿœÂ°…D…¤…,hk!
__sig__5kÔkìkl!D…¬…ûÿÿÿLmm!``f1 and f2 are considered as homogeneous polynomials of degree n. It is checked whether they belong to the same GL(2,k)-orbit, where k is the coefficient field, modulo multiplication by scalars. As a second value, a sequence of matrix entries [a,b,c,d] is returned such that f2(x) = const * (c*x + d)^n * f1((a*x + b)/(c*x + d)). Verbose flag: CrvHypIso}à}H~o!
__sig__6`xp!D…¼…ûÿÿÿˆ‚À‚¬„¯ EvaluateEquationQüQHR° ÿÿÿÿÿÿÿÿ°‚± EEvaluates the homogeneous equation of C at (a,b,c) where s = [a,b,c].˜X² ÿþÿgþÿûÿûÿûÿûÿÐ‚¨Ä‚$]³ 
	__sig__15T_l_´ ¨Ì‚ûÿÿÿ,aµ  eReturns true if and only if S specifes a point on C, and, if so, returns the point as a second value.te¶ ghûÿûÿûÿ<„à‚¨Ô‚œj· 
	__sig__16ll¸ ¨Ü‚ûÿÿÿ|m¹ )Apply the hyperelliptic involution to P.  pèpº hhûÿûÿûÿûÿð‚¨ä‚v» 
	__sig__17Xwpw¼ ¨ì‚ûÿÿÿøx½ )Apply the hyperelliptic involution to P. 4{T{¾ hhûÿûÿûÿûÿŒ„ƒ¨ô‚€!$sFind all points on J with first component a and degree d. So far only for genus 2 and curve of the form y^2 = f(x).Fq;!0ûÿûÿûÿûÿÜŠ„„˜ƒx„d !
	__sig__16Y8!!˜ƒ€„ûÿÿÿB€Y8"!?Returns the divisor class [P1 - P2] as a point on the Jacobian.À#!hhûÿûÿûÿûÿ‹”„˜ƒˆ„$!
	__sig__17Ô%!˜ƒ„ûÿÿÿ&!
,'!ÿhÿhûÿûÿûÿûÿ´„¤„˜ƒ˜„(!
	__sig__18ˆ)!˜ƒ „ûÿÿÿ*!
JacobianPoint+!ÿÿÿÿÿÿÿÿœ„,!IThe difference of the two points in the Jacobian (for internal use only).-!hhûÿûÿûÿûÿ¼„˜ƒ°„P.!
	__sig__19d/!˜ƒ¸„ûÿÿÿ0!(€The Euler factor of a Jacobian over a finite field, i.e. the reciprocal characteristic polynomial of Frobenius acting on H^1(J).…_;1!ûÿûÿûÿûÿÔ„Ì„˜ƒÀ„õÂÂ:2!
	__sig__20„3!˜ƒÈ„ûÿÿÿ4!]The Euler factor of the base extension of J to K, where the base field of J is the rationals.,5!qûÿûÿûÿûÿÜ„˜ƒÐ„6!
	__sig__21Ð7!˜ƒØ„ûÿÿÿ8!BThe number of rational points on the Jacobian over a finite field.9!ûÿûÿûÿûÿ¼Øì„˜ƒà„ˆ:!
	__sig__22d;!˜ƒè„ûÿÿÿ<!VThe order of the group of rational points of J, over a finite field, in factored form.=!ûÿûÿûÿûÿü„˜ƒð„>!
	__sig__23?!˜ƒø„ûÿÿÿ)@!*‡A bound on the size of the rational torsion subgroup by looking at the first n good primes, where the base field of J is the rationals.¤*A!ûÿûÿûÿûÿØÑ…˜ƒ…B!
	__sig__24T,C!˜ƒ…ûÿÿÿÄ!
__sig__2\„Å!ä†‡ûÿÿÿ˜ÑXûÿûÿûÿûÿ$‡L3ä†H3˜Ç!ÿÿÿÿÿÿÿÿD3È!"mGiven a prime number p, return a function that decides whether its argument (an integer) is a square in Z_p. ,É!Xûÿûÿûÿûÿ4‡,‡ä† ‡¼Ê!
__sig__5ÔìË!ä†(‡ûÿÿÿ|Ì!&}Given a prime ideal pZ in the integers, return a function that decides whether its argument (an integer) is a square in Z_p. èÍ!Xûÿûÿûÿûÿ<‡ä†0‡˜!Î!
__sig__6"°"H#Ï!ä†8‡ûÿÿÿ¸$Ð!=Compute the Newton Polygon of f with respect to the prime p. ¤(Ñ!ûÿûÿûÿûÿT‡L‡ä†@‡¼.Ò!
__sig__70Ü01Ó!ä†H‡ûÿÿÿÌ4Ô!ECompute the Newton Polygon of f with respect to the prime ideal pid. Ü9Õ!ûÿûÿûÿûÿÃ\‡ä†P‡¼<Ö!
__sig__8=ô=>×!ä†X‡ûÿÿÿ¤?Ø!2«Tests whether the curve y^2 = f(x), where f has coefficients coercible into O = Order(pid), has a point with x-coordinate in the p-adic ring given by completing O at pid. ¼HôHÙ!ûÿûÿûÿûÿt‡l‡ä†`‡NÚ!
__sig__9O€O˜OÛ!ä†h‡ûÿÿÿ`QÜ!&}Tests whether the curve y^2 = f(x), where f has integral coefficients, has a point with x-coordinate in the p-adic integers. WÝ!ûÿûÿûÿûÿ|‡ä†p‡Ì[Þ!
	__sig__10H]p]ß!ä†x‡ûÿÿÿH`à!4°Tests whether C (a curve of genus 2 over the rationals) is deficient at p (p = 0 stands for the infinite place). This means that there is no point of odd degree over Q_p on C. fˆf fá!gûÿûÿûÿûÿŒ‡ä†€‡Hlâ!
	__sig__11`mxmã!ä†ˆ‡ûÿÿÿ¼oä!
IsDeficientÜrså!ÿÿÿÿÿÿÿÿ„‡æ!:ÇDetermines if J, the Jacobian of a curve (of genus two over the rationals), is even in the sense of Poonen-Stoll, i.e. if its Shafarevich-Tate group (supposed to be finite) has square order, or not. D~\~t~ç!ûÿûÿûÿûÿdÙ¤‡ä†˜‡Ä…n!ûÿûÿûÿÀ…D…´…¬q!
IsGL2Equivalent$r!ÿÿÿÿÿÿÿÿ¸…s!./usr/local/Magma2.8/package/SchCrvHyp/models.m	¨
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CrvHypMinimal\v!,Returns true if C has integral coefficients.w!gûÿûÿûÿûÿä…Ð…Ø…<x!
__sig__0<\y!Ð…à…ûÿÿÿ¼z!7Returns true if C has p-adically integral coefficients.TlŒ{!gûÿûÿûÿûÿô…Ð…è…"|!
__sig__1#¨#À#}!Ð…ð…ûÿÿÿ@%~!
IspIntegralX'x'!ÿÿÿÿÿÿÿÿì…€!5Returns true if C is normal over the p-adic integers.l/!gûÿûÿûÿûÿ†Ð…†\6‚!
__sig__278l8ƒ!Ð…†ûÿÿÿL:„!
	IspNormalT;|;…!ÿÿÿÿÿÿÿÿ††!NReturns false, false if C is not a p-integral p-minimal model. Returns true, false if C is p-integral, p-minimal, but not the unique p-minimal model (up to the action of GL(2,Z_p)). Returns true, true if C is the unique (up to the action of GL(2,Z_p)) p-integral p-minimal model.J\JtJ‡!gûÿûÿûÿ$†Ð…†˜Oˆ!
__sig__3PQ0Q‰!Ð… †ûÿÿÿ¤R!HReturns an isomorphic curve C' given by polynomials with integral coefficients, together with the map C -> C' and the transformation used. Base field must be the rationals. If Reduce is set, common divisors of the coefficients are eliminated as far as possible.h_‘!gg–ûÿûÿûÿL†Ð…@†xb’!
__sig__5c@d°d“!Ð…H†ûÿÿÿàe”!.˜Checks whether C has a model of the form y^2 = f(x) with f of odd degree. If so, returns this model together with the map from the old to the new model.mm¨m•!gg–ûÿûÿ\†Ð…P†4t–!
__sig__6u\utu—!Ð…X†ûÿÿÿØv˜!HasOddDegreeModel@xXx™!ÿÿÿÿÿÿÿÿT†š!*ŽGiven a hyperellitpic curve C over the rationals and a prime p, this returns a normal Weierstrass model of the curve over the p-adic integers.8…D!
TorsionBoundfgE!ÿÿÿÿÿÿÿÿ…F!&xThe order of the point on the Jacobian over a finite field or the rationals. Result is zero if point has infinite order.rÜrsG!ûÿûÿûÿûÿ<$…˜ƒ…€vH!
	__sig__25¼wÔwI!˜ƒ …ûÿÿÿxyJ!"nReturns true iff n is the order of the Jacobian point P. Maybe expensive over infinite fields when n is large.tL!
	__sig__264„¿ 
	__sig__18À ¨ü‚ûÿÿÿÁ )Compute the discriminant of the curve C. Â gþÿûÿûÿûÿûÿ,Æƒ¨ƒÃ 
	__sig__19Ä ¨ƒûÿÿÿÅ `eReturns the sequence of primes where the given model of C has bad reduction. C must be defined over a number field, and the defining polynomials must have integral coefficients. If C is defined over the rationals, the parameter Badness can be given; then only primes are returned such that the valuation of the discriminant at the prime is at least Badness.Æ gûÿûÿûÿûÿÌƒ ƒ¨ƒÇ 
	__sig__20È ¨ƒûÿÿÿÉ  dReturns the indexed set of all rational points on the hyperelliptic curve C with given x-coordinate.Ê þÿg0ûÿûÿûÿûÿHƒ0ƒ¨$ƒË 
	__sig__21Ì ¨,ƒûÿÿÿÍ  dReturns the indexed set of all rational points on the hyperelliptic curve C with given x-coordinate.Î þÿg0ûÿûÿûÿûÿXƒ@ƒ¨4ƒ¯"ÿþÿ ûÿûÿûÿÄŠ<Š¸Š$±"<ŠÀŠûÿÿÿ<²"NReturn the indexed set of points on K with first three coordinates given by s.p³"ÿþÿ0ûÿûÿûÿûÿìŠÔŠ<ŠÈŠ¤´"
__sig__6¸µ"<ŠÐŠûÿÿÿÐ¶"NReturn the indexed set of points on K with first three coordinates given by s.œ·"ÿþÿ0ûÿûÿûÿûÿüŠäŠ<ŠØŠÐ¸"
__sig__7ä¹"<ŠàŠûÿÿÿüº"JReturn the indexed set of points on J mapping to P on Kummer surface of J.Ì	»" 0ûÿûÿûÿûÿ\¡ôŠ<ŠèŠ
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¾"JReturn the indexed set of points on J mapping to P on Kummer surface of J.Ô¿" 0ûÿûÿûÿûÿ‹<ŠøŠ„!À"
__sig__9$Ø$Á"<Š‹ûÿÿÿ`5Â"-Returns the negation of P, equal to P itself.´BÃ"  ûÿûÿûÿûÿ¼Æ‹<Š‹ŒKÄ"
	__sig__10 KÅ"<Š‹ûÿÿÿ¸KÆ""Returns the double of the point P.(RÇ"  ûÿûÿûÿûÿ$‹<Š‹ ^È"
	__sig__11´^É"<Š ‹ûÿÿÿÌ^Ë"ÿÿÿÿÿÿÿÿ‹Ì" eGiven the images on the Kummer surface of points P, Q, P-Q on the Jacobian, returns the image of P+Q.<gÍ"    ûÿûÿûÿûÿ<‹<Š0‹¤ƒÎ"
	__sig__12È‡Ï"<Š8‹ûÿÿÿ@Šþ!Autoø!

Polynomialb¨bù!Scalar°dö!
ú!The set of automorphisms of C.gû!g”ûÿûÿûÿûÿô‡Ä‡è‡Èlü!
__sig__0mÀmØmý!Ä‡ð‡ûÿÿÿ„pÿ!ÿÿÿÿÿÿÿÿì‡"&The set of isomorphisms from C1 to C2.€v"gg”ûÿûÿûÿûÿˆÄ‡ˆz"
__sig__1{0{P{"Ä‡ˆûÿÿÿt~"Isomt"ÿÿÿÿÿÿÿÿˆ›!gg–ûÿûÿûÿt†Ð…h†¬œ!
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omCrvHyp>"Returns true if f is in M.Ld?"”ûÿûÿûÿûÿ‰Ä‡øˆà@"
	__sig__14èA"Ä‡‰ûÿÿÿ€B"HackobjInHomCrvHypØðC"ÿÿÿÿÿÿÿÿüˆL"
	__sig__16vHypàD"The parent of f.pE"–”ûÿûÿûÿûÿ‰Ä‡‰"F"
	__sig__15¬#Ä#G"Ä‡‰ûÿÿÿD%H"HackobjParentMapCrvHyp(I"ÿÿÿÿÿÿÿÿ‰J"The domain of M.-è-.K"”gûÿûÿûÿûÿL‰4‰Ä‡(‰Ð4M"Ä‡0‰ûÿÿÿ7N"The codomain of M.È9à9O"”gûÿûÿûÿûÿ\‰D‰Ä‡8‰À<P"
	__sig__17ø= >Q"Ä‡@‰ûÿÿÿ¨?R"The domain of f.AˆAÀAS"–gûÿûÿûÿûÿpØT‰Ä‡H‰HFT"
	__sig__18hH¨HU"Ä‡P‰ûÿÿÿxJV"The codomain of f.lMŒMW"–gûÿûÿûÿûÿ€Ød‰Ä‡X‰dQX"
	__sig__19R¨RY"Ä‡`‰ûÿÿÿ@T["–ûÿûÿûÿûÿÄ–t‰Ä‡h‰ÐX\"
	__sig__20¸Z[]"Ä‡p‰ûÿÿÿ´\°%
assPlace^"
4`L`_"–þÿûÿûÿ\¢„‰Ä‡x‰Üb`"
	__sig__21üdea"Ä‡€‰ûÿÿÿDfb"!The inverse of the isomorphism f.h,hc"––ûÿûÿûÿûÿ”‰Ä‡ˆ‰¬md"
	__sig__22¨oÀoe"Ä‡‰ûÿÿÿhsf".The composition of the isomorphisms f1 and f2.Øug"–––ûÿûÿûÿûÿL‹¤‰Ä‡˜‰Lyh"
	__sig__23pz˜zi"Ä‡ ‰ûÿÿÿh}j"7The n-fold composition of the automorphism with itself.€°€È€l"
	__sig__24…È…ž!
pNormalModel$Ÿ!ÿÿÿÿÿÿÿÿl†®!
	__sig__10Model !*ŽGiven a hyperelliptic curve C over the rationals and a prime p, returns a minimal Weierstrass model of C with respect to the p-adic valuation. ¡!gg–ûÿûÿûÿŒ†Ð…€†à¢!
__sig__8Øð£!Ð…ˆ†ûÿÿÿL¤!pMinimalWeierstrassModel´¥!ÿÿÿÿÿÿÿÿ„†¦!HGiven a hyperelliptic Curve C over the rationals, this returns a globally minimal Weierstrass model of C. If Bound is set, it gives an upper bound for the bad primes that are checked (this uses trial division, so Bound should not be much larger than 10^7.)./p/§!gg–ûÿûÿûÿ¤†Ð…˜†`6¨!
__sig__978p8©!Ð… †ûÿÿÿP:ª!MinimalWeierstrassModelÐ;«!ÿÿÿÿÿÿÿÿœ†¬!$qGiven a curve with integral coefficients, computes a model that is reduced with respect to the action of SL(2,Z).4C¼C­!gg–ûÿûÿûÿ¼†Ð…°†(H¯!Ð…¸†ûÿÿÿJ°!
ReducedModelLM±!ÿÿÿÿÿÿÿÿ´†¼!__sig__0erstrassModel4Q²!2¬Given a hyperelliptic Curve C over the rationals, returns a globally minimal Weierstrass model of C that is reduced with respect to the action of SL(2,Z). (Compare Reduce.)XÐX³!gg–ûÿûÿûÿÔ†Ð…È†t]´!
	__sig__11ä_ü_µ!Ð…Ð†ûÿÿÿ\a¶!ReducedMinimalWeierstrassModel¬b·!ÿÿÿÿÿÿÿÿÌ†¸!./usr/local/Magma2.8/package/SchCrvHyp/selmer.m	hÔ
	¹!./usr/local/Magma2.8/package/SchCrvHyp/selmer.m	hÔ
	»!ûÿûÿûÿûÿô†ä†è†(p½!ä†ð†ûÿÿÿ¨sò!Domainata¾!
ÀuØu¿!ûÿûÿûÿûÿ‡ä†ø†LyÀ!
__sig__1zpz˜zÁ!ä†‡ûÿÿÿh}Â!
~°~Ã!“ûÿûÿûÿ‡ä†‡†,†d†Š!

IspMinimals¨s‹!ÿÿÿÿÿÿÿÿ†Œ!RReturns an isomorphic curve C' in the form v^2 = f(u) and the isomorphism C -> C'.dy|y!gg–ûÿûÿûÿ<†Ð…0†x~Ž!
__sig__4°!Ð…8†ûÿÿÿ¬„Ð"
	PseudoAdd\„Ñ"ÿÿÿÿÿÿÿÿ4‹Ò"/The n-th multiple of P on the Kummer surface K.ÔhÓ"  ûÿûÿûÿûÿ\‹T‹<ŠH‹ðÔ"
	__sig__13\tÕ"<ŠP‹ûÿÿÿÀÖ"/The n-th multiple of P on the Kummer surface K.|”¬×"  ûÿûÿûÿûÿŒÆd‹<ŠX‹¼Ø"
	__sig__14ÔìÙ"<Š`‹ûÿÿÿDÚ""gGiven the images on the Kummer surface of points P, Q, P-Q on the Jacobian, returns the image of P+n*Q.`ØÛ"    ûÿûÿûÿûÿt‹<Šh‹#Ü"
	__sig__15 $¸$Ý"<Šp‹ûÿÿÿ'Þ"PseudoAddMultiple¼()ß"ÿÿÿÿÿÿÿÿl‹á"=Extends the base field of the Kummer surface K by applying j.
	øÚ
	à"ûÿûÿûÿûÿ‹ˆ‹<Š„‹PÛ
	ã"
	__sig__20øÚ
	PÛ
	å"PExtends the finite base field of the Kummer surface K to its degree n extension.ace K by apä"ûÿûÿûÿûÿŒ£˜‹<Š”‹¤‹ç"
	__sig__21ç	__siæ"<Šœ‹ûÿÿÿûÿÿÿè"The defining equation of K. surfaceé"KûÿûÿûÿûÿŒ¬‹<Š ‹ê"
	__sig__25ç	__sië"<Š¨‹ûÿÿÿûÿÿÿò"
	__sig__23¨‹ûÿÿÿì"4Extends the base field of the Kummer surface K to F.V\Ví"ýÿûÿûÿûÿûÿ€‹¼‹<Š°‹T[î"
	__sig__19ˆ\°\ï"<Š¸‹ûÿÿÿh_â"<ŠŒ‹ûÿÿÿ(að"(The base field of the Kummer surface K. bÀbØbñ"õÿûÿûÿûÿûÿßÌ‹<ŠÀ‹ fó"<ŠÈ‹ûÿÿÿ(hô"(The base field of the Kummer surface K. l°lÈlõ"õÿûÿûÿûÿûÿ°½Ü‹<ŠÐ‹Ärö"
	__sig__24|tœt÷"<ŠØ‹ûÿÿÿÔuø"//usr/local/Magma2.8/package/SchCrvHyp/torsion.mPâ
	ù"//usr/local/Magma2.8/package/SchCrvHyp/torsion.mPâ
	ú"OThe rational 2-torsion subgroup of J for curves of genus 2 in simplified model.tŒ¬û"“ûÿûÿûÿô‹ä‹è‹‡ü"
__sig__0ˆ‰ ‰k"––ûÿûÿûÿûÿÉ´‰Ä‡¨‰Œ
m"Ä‡°‰ûÿÿÿÔn"The image of P under f.`xo"í–íûÿûÿûÿûÿÜ‰Ä‰Ä‡¸‰Äp"
	__sig__25¨Àq"Ä‡À‰ûÿÿÿìr"The image of P under f.4Ts"–ííûÿûÿûÿûÿì‰Ô‰Ä‡È‰|t"
	__sig__26t”u"Ä‡Ð‰ûÿÿÿìv"The image of P under f.L`€w"–ûÿûÿûÿûÿô”ä‰Ä‡Ø‰,!x"
	__sig__27€"˜"y"Ä‡à‰ûÿÿÿh$z"The image of P under f.Ð&è&'{"–ûÿûÿûÿûÿô‰Ä‡è‰¬,|"
	__sig__28d.¼.}"Ä‡ð‰ûÿÿÿD1~"The preimage of P under f.6,6"í–íûÿûÿûÿûÿŠŠÄ‡ø‰œ:€"
	__sig__29¤;Ì;"Ä‡Šûÿÿÿì<‚"The inverse image of P under f.Ü>,?D?ƒ"–íûÿûÿûÿûÿ,ŠŠÄ‡ŠD„"
	__sig__30œE´E…"Ä‡Šûÿÿÿ\G†"The preimage of P under f.üIJ‡"–ûÿûÿûÿûÿL©$ŠÄ‡ŠXOˆ"
	__sig__31¨PàP‰"Ä‡ ŠûÿÿÿtRŠ"The inverse image of P under f.<TTT|T‹"–ûÿûÿûÿûÿ4ŠÄ‡(Š,YŒ"
	__sig__32<[T["Ä‡0Šûÿÿÿ ]Ž"./usr/local/Magma2.8/package/SchCrvHyp/kummer.m	pî
	",./usr/local/Magma2.8/package/SchCrvHyp/kummer.m	pî
	"
Equationc@d°d’"
	DEquation˜e°e“"Deltapf”"
BBMatrixg°gÈgš"
__sig__0•"
AuxPolynomials˜l–"
NormalizePoint¨m‘"—"SelectCoordinatesŒs¤s˜"AReturns the Kummer surface of the Jacobian J of a genus 2 curve. ˜vØv™"ûÿûÿûÿûÿlŠ<Š`ŠTz›"<ŠhŠûÿÿÿÀ|# Vûÿûÿûÿûÿ€ŒxŒLŒlŒL#LŒtŒûÿÿÿ(#*‹Computes the logarithmic naive height of P given by the mapping J -> K -> P^3. J must be a Jacobian of a genus 2 curve over the rationals. 0H #VûÿûÿûÿûÿˆŒLŒ|Œð"#LŒ„Œûÿÿÿ ## cComputes the canonical height of P. The genus must be 2, and the base field must be the rationals. Øð$#Vûÿûÿûÿûÿ Œ˜ŒLŒŒŒà%#
__sig__3X&#LŒ”Œûÿÿÿ'#JComputes the canonical height of P. The base field must be the rationals. ¬#Ä#(# Vûÿûÿûÿûÿ¨ŒLŒœŒH()#
__sig__4* *À**#LŒ¤Œûÿÿÿð,+#4¶Computes the canonical height pairing of P and Q (on the same Jacobian), defined by <P,Q> := (h(P+Q) - h(P) - h(Q))/2. The genus must be 2, and the base field must be the rationals.  9,#Vûÿûÿûÿûÿ¸ŒLŒ¬Œ<-#
__sig__5=¨=Ð=.#LŒ´Œûÿÿÿx?/#2®Given a sequence of points P_i on a Jacobian (of a curve of genus two over the rationals), this returns the matrix (<P_i, P_j>), where < , > is the canonical height pairing. ¨H0#ÿûÿûÿûÿûÿÈŒLŒ¼ŒÄM1#
__sig__6N4O\O2#LŒÄŒûÿÿÿ4Q3#dvGiven a sequence of points on the Jacobian of a genus two curve over the rationals), this computes the regulator, i.e. the determinant of the height pairing matrix. It is zero when the points are dependent in the Mordell-Weil group; otherwise it is the square of the volume of the parallelotope spanned by the points (in J(Q) tensor R, with the canonical height as metrics).Œa4#ÿVûÿûÿûÿûÿØŒLŒÌŒDe5#
__sig__7f,fDf6#LŒÔŒûÿÿÿÌg7#0/usr/local/Magma2.8/package/SchCrvHyp/chabauty.mlX÷
	ú
	8#0/usr/local/Magma2.8/package/SchCrvHyp/chabauty.mo¨÷
	ú
	9#
JacHypChabautys:#&{Returns the rational points on the curve of which J is the Jacobian, under the condition that J has Mordell-Weil rank zero.ðw x;#0ûÿûÿûÿûÿôŒàŒèŒ¬{<#
__sig__0~`~x~=#àŒðŒûÿÿÿà>#
	Chabauty0À‚¬„?#ÿÿÿÿÿÿÿÿìŒB#
__sig__1‰tŠ,‹œ"
KummerSurface°"ÿÿÿÿÿÿÿÿdŠž"1The point on K specified by the coordinates of P.‰$‰S$6/usr/local/Magma2.8/package/CrvMod/class_polynomials.m	0û
	ë0IsFull\
$ggûÿûÿûÿûÿ0$\$
ReduceCurvep$ÿÿÿÿÿÿÿÿ(l$ ‘¬‘ûÿÿÿe/ge$3/usr/local/Magma2.8/package/CrvMod/modular_curves.m@ü
	¸	$3/usr/local/Magma2.8/package/CrvMod/modular_curves.mü
	¸	$
ModularCurve¬$0The canonical involution of the modular curve X.Ô$%ûÿûÿûÿûÿXDL¤	$
__sig__0	¸	$DTûÿÿÿÐ	$CanonicalInvolutionè	$ÿÿÿÿÿÿÿÿP!$
__sig__2tion
$3The Atkin-Lehner involution of the modular curve X.D
$%ûÿûÿûÿûÿpDdè $
__sig__1!x!$DlûÿÿÿÔ$$AtkinLehnerInvolution\5$ÿÿÿÿÿÿÿÿh$ZThe base curve X(1) of the modular curve X_0(N), followed by the projection to this curve.ŒK $%%ûÿûÿûÿˆD|ÀK"$D„ûÿÿÿ(R#$
	BaseCurve€^$$ÿÿÿÿÿÿÿÿ€n$ÿÿÿÿÿÿÿÿ¨‘%$6/usr/local/Magma2.8/package/CrvMod/modular_equations.m	¸	&$6/usr/local/Magma2.8/package/CrvMod/modular_equations.m	¸	'$,The Atkin modular equation of prime level p.¨v($KûÿûÿûÿûÿLõ¨˜œ°ƒ)$
__sig__0‡Ü‡*$˜¤ûÿÿÿPŠ&%MinorsreeŸ"  ûÿûÿûÿŒŠ„Š<ŠxŠ”V "
__sig__1XLX”X¡"<Š€Šûÿÿÿ[¢"8The point on K specified by the sequence of coordinates.^ _8_£"ÿþÿ ûÿûÿûÿœŠ”Š<ŠˆŠHb¤"
__sig__2cc¨c¥"<ŠŠûÿÿÿ°e¦"!The origin on K, where zero is 0.Pgˆg§" ûÿûÿûÿ¬Š¤Š<Š˜Šm¨"
__sig__3mnn©"<Š Šûÿÿÿäpª"The image of P on K.tœt«" ûÿûÿûÿ´Š<Š¨Š¤w¬"
__sig__4xÌxôx­"<Š°Šûÿÿÿ”z®"&zCheck if point specified by the homogeneous coordinates s is on K. If so, return corresponding SrfKumPt as a second value.‚„‚°"
__sig__5†(†`†Ê"Double¸‡è!
	__sig__12pé!ä† ‡ûÿÿÿŒ
ê!æwCompute the 2-Selmer group of J (a Jacobian of a hyperelliptic curve defined over the rationals). The first value is the upper bound for the Mordell-Weil rank deduced from the computation, the second value is the dimension of the Selmer group. The third value gives a presentation of the (fake) Selmer group (but is only defined when UseUnits is true). It is a triple <seq1, seq2, seq3>, where seq1 is a sequence of elements in L = Q[x]/(f(x)) representing a basis of some subgroup S1 of L^*/(L^*)^2, seq2 and seq3 are sequences of integers representing bases of subgroups S2 and S3 of Q^*/(Q^*)^2, respectively, and the (fake) Selmer group S fits into the exact sequence 0 --> S3 --> S2 --> S1 --> S --> 0. This data is in terms of the possibly modified model of the curve that the algorithm actually uses. As a fourth value, a set of the number fields used in the function is returned.¼*Ô*ì*ë!Åûÿ´‡ä†¨‡´0ì!
	__sig__13l2¼3í!ä†°‡ûÿÿÿŒ6î!TwoSelmerGroupDataä89ï!ÿÿÿÿÿÿÿÿ¬‡è#./usr/local/Magma2.8/package/SchCrvHyp/mestre.m=ð!4/usr/local/Magma2.8/package/SchCrvHyp/isomorphisms.m	`	ñ!B4/usr/local/Magma2.8/package/SchCrvHyp/isomorphisms.m	`	ó!õ!Parentyp÷!Matrix$H"The identity endomorphism of C.tJŒJK"”–ûÿûÿûÿûÿØè$ˆÄ‡ˆèO"
__sig__2QHQ`Q	"Ä‡ ˆûÿÿÿìR6"

ntMapCrvHyp<T
"6The hyperelliptic involution on C as a transformation.¤W"g–ûÿûÿûÿûÿ4ˆÄ‡(ˆP\"
__sig__3]^@^"Ä‡0ˆûÿÿÿ¸`"HyperellipticInvolutionèa"ÿÿÿÿÿÿÿÿ,ˆ".The transformation of C given by the matrix sequence t, the scaling factor e, and the polynomial u, returning the codomain curve followed by the isomorphism.Hl"þÿÿþÿgg–ûÿûÿûÿTˆLˆÄ‡@ˆ„p"
__sig__4sŒs¤s"Ä‡Hˆûÿÿÿtu".The transformation of C given by the matrix sequence t, the scaling factor e, and the polynomial u, returning the codomain curve followed by the isomorphism.P{"ÿþÿgg–ûÿûÿûÿdˆ\ˆÄ‡Pˆ€"
__sig__5„…8…"Ä‡Xˆûÿÿÿ‡>+ÿÿÿÿÿÿÿÿè¬?+EThe element of the K3 database DB with codimension c and AFR number n@@+bûÿûÿûÿûÿ­Ì«ü¬tA+
	__sig__11ˆB+Ì«­ûÿÿÿ C+K3SurfaceFromAFR¸D+ÿÿÿÿÿÿÿÿ­E+3The element of the K3 database DB with identifier i˜F+bûÿûÿûÿûÿ ­Ì«­ÌG+
	__sig__12àH+Ì«­ûÿÿÿøI+AA record of the attributes of V in the format of a database entryÄ	J+b‹ûÿûÿûÿûÿ0­Ì«$­ø	K+
	__sig__13
L+Ì«,­ûÿÿÿ$
M+
NumericalRecord8
N+ÿÿÿÿÿÿÿÿ(­O+The codimension of XèP+bûÿûÿûÿûÿH­Ì«<­Ô$Q+
	__sig__14H5R+Ì«D­ûÿÿÿp5S+<The number g associated to the polarising linear system on X€DT+bûÿûÿûÿûÿ`èX­Ì«L­¤KU+
	__sig__15¸KV+Ì«T­ûÿÿÿRW+ cThe Hilbert series of the scheme X after cutting by hyperplane sections of degrees the weights of X¤^X+bûÿûÿûÿûÿh­Ì«\­Ø^Y+
	__sig__16ì^Z+Ì«d­ûÿÿÿ_[+HilbertNumeratorg,g\+ÿÿÿÿÿÿÿÿ`­]+QThe weights of variables of X before unprojection conditions have been consideredÈ‡^+bûÿûÿûÿûÿ€­Ì«t­ˆ_+
	__sig__17Øbm$
BaerDerivationkage/ä#
AlphaDivPolˆf fè 
$á#</usr/local/Magma2.8/package/SchCrvHyp/Genus2/formalarithg2.m	à 	é#
./usr/local/Magma2.8/package/SchCrvHyp/mestre.m	Ø	Hi
	ä»çæ#=/usr/local/Magma2.8/package/SchCrvHyp/Genus2/twotorsionlift.m	à 	ü#
__sig__3smsmIgusaê#WBuild a curve of genus 2 with the given IgusaClebsch invariants by Mestre's algorithm. P{x{¨{ë#gûÿûÿûÿûÿ´¤¨Ä€ì#
__sig__0…Œ…Ä…í#¤°ûÿÿÿ¸‡î#HyperellipticCurveFromIgusa ŒdŒc#NGiven a degree d binary form F(x,z) with real coefficients, represented by pol = F(x,1) of degree d such that F is squarefree, returns a GL_2(Z)-equivalent form (in dehomogenized form) and the transformation matrix (an element of GL_2(Z)). If Simple is set, Q_0-reduction is used.,Dd#kûÿûÿûÿ˜LŒìe#
__sig__3ÄÜf#L”ûÿÿÿ<g#0Tests whether its argument is squarefre or not. œ¼h#ûÿûÿûÿûÿ¨LœŒi#
__sig__4Èèj#L¤ûÿÿÿØk#2/usr/local/Magma2.8/package/SchCrvHyp/jac_mutate.mè#	P-	l#2/usr/local/Magma2.8/package/SchCrvHyp/jac_mutate.m8$	P-	m#Change pt1 to pt1 + pt2. $(D(n#ûÿûÿûÿûÿûÿÀ°´ü-o#
__sig__0/40l0p#°¼ûÿÿÿ¼3q#Change pt1 to pt1 - pt2. ä6,7r#ûÿûÿûÿûÿûÿÐ°Ä,;s#
__sig__1<D<\<t#°ÌûÿÿÿÌ=u#Change pt to n*pt. \?t?v#ûÿûÿûÿûÿûÿà°ÔÄDw#
__sig__2EÌEäEx#°Üûÿÿÿ$Hy#-/usr/local/Magma2.8/package/SchCrvHyp/igusa.m	P-	z# -/usr/local/Magma2.8/package/SchCrvHyp/igusa.m	P-	{#0¥Given a polynomial of degree at most 6, compute the Clebsch invariants A, B, C, D (as on p. 317 of Mestre). The base field must not be characteristic not 2, 3, or 5.V|#ûÿûÿûÿûÿŽøèì[}#
__sig__0\8\P\~#èôûÿÿÿ8_#ClebschInvariantsÐ`è`€#ÿÿÿÿÿÿÿÿð#"lCompute the Clebsch invariants of a genus 2 curve. The base field must not be characteristic not 2, 3, or 5.fg‚#gûÿûÿûÿûÿŽèŽÈlƒ#
__sig__1mÀmØm„#èŽûÿÿÿ„p‘#
__sig__4ants¤s…#FùCompute the Igusa J-invariants of the curve y^2 + h*y - f = 0, scaled by [16, 16^2, 16^3, 16^4, 16^5]. The polynomial h must have degree at most 3, the polynomial f must have degree at most 6, and the characteristic of the base ring should not be 2.\~t~†#ûÿûÿûÿûÿ0Ž ŽèŽÄ…‡#
__sig__2†Ø†‡ˆ#èŽûÿÿÿ(‹v%
nctionsJ#
FindPowerSeriesòì];K#ÿÿÿÿÿÿÿÿT#
CrvHypReducebautyL#For internal use only. Ì…M#þÿÿ#ûÿûÿûÿûÿ<àŒ0à‹N#
__sig__3D%
eMatrixný"ä‹ð‹ûÿÿÿ˜þ"<ÔFinds the rational torsion subgroup of J. The curve of J must have genus 2 and be defined over the rationals and have form y^2 = f(x) with integral coefficients. Second value gives the map from the group into J. tÿ"“ûÿûÿûÿŒä‹ø‹#
__sig__1ô#ä‹Œûÿÿÿl#//usr/local/Magma2.8/package/SchCrvHyp/descent.mø:	#//usr/local/Magma2.8/package/SchCrvHyp/descent.mø:	#(‚Computes a reduced basis (with respect to the height pairing) of the subgroup in J(Q)/torsion generated by the given points on J. Ð ü #ÿûÿûÿûÿ,ŒŒŒŒ¸$#
__sig__0&ˆ&Ð&#ŒŒûÿÿÿ¤(‰#ScaledIgusaInvariantsÔŠ#ÿÿÿÿÿÿÿÿŽ‹#0 Compute the Igusa J-invariants of a polynomial of degree at most 6, scaled by [16, 16^2, 16^3, 16^4, 16^5]. The coefficient ring must not have characteristic 2.ˆ°Œ#ûÿûÿûÿûÿ8Žè,ŽŒ#
__sig__3d|Ž#è4Žûÿÿÿ¼#.œCompute the Igusa J-invariants of the curve y^2 + h*y - f = 0. The polynomial h must have degree at most 3, and the polynomial f must have degree at most 6.#ûÿûÿûÿûÿXŽHŽè<ŽØ’#èDŽûÿÿÿ,!“#
IgusaInvariants˜"”#ÿÿÿÿÿÿÿÿ@Ž•#6½Compute the Igusa J-invariants of a polynomial of degree at most 6. The integer 2 must be a unit of the coefficient ring. If Quick, the base field must not be of characteristic 2, 3, or 5. ´0–#ûÿûÿûÿûÿhŽ`ŽèTŽŒ7—#
__sig__59d9œ9˜#è\Žûÿÿÿü:™#(‚Compute the Igusa J-invariants of a genus-2 curve over a field. If Quick, the base field must not be of characteristic 2, 3, or 5.¼?Ô?š#gûÿûÿûÿûÿpŽèdŽDE›#
__sig__6F,FDFœ#èlŽûÿÿÿôH#.œCompute the Igusa J-invariants of the curve y^2 + h*y - f = 0. The polynomial h must have degree at most 3, and the polynomial f must have degree at most 6.Q0Qž#ûÿûÿûÿûÿŽ€ŽètŽ|TŸ#
__sig__7V<V\V #è|Žûÿÿÿ”X¥#
__sig__8ZtZœZ¦#è”ŽûÿÿÿP\§#(‚Compute the Igusa J-invariants of a genus-2 curve over a field. If Quick, the base field must not be of characteristic 2, 3, or 5.`bxb¨#gûÿûÿûÿûÿ¨ŽèœŽ@f©#
__sig__9gPgˆgª#è¤Žûÿÿÿˆk»#
	__sig__13ants˜l«#BèCompute the Igusa-Clebsch invariants A', B', C', D' of a polynomial of degree at most 6 (see p. 319 of Mestre). The integer 2 must be a unit of the coefficient ring. If Quick, the base field must not be of characteristic 2, 3, or 5.w¼wÔw¬#ûÿûÿûÿûÿÈŽ¸Žè¬ŽP{­#
	__sig__10Ü}D~®#è´Žûÿÿÿ¬¯#IgusaClebschInvariants¨„°#ÿÿÿÿÿÿÿÿ°Ž³#
	__sig__11pŠ(‹@#n™Returns an indexed set of triples <x,z,v,k> such that there are at most k pairs of rational points on C with x-coordinate mapping to x/z mod p^v, and such that and there are no non-Weierstrass points outside the union of these classes. The algorithm uses Chabauty's method, which applies to curves of genus 2 with Jacobian of rank 1, for which ptJ must be a generator, and p must be a prime of good reduction.d|A#0ûÿûÿûÿûÿàŒ|C#àŒûÿÿÿìD#
ChabautyôE#ÿÿÿÿÿÿÿÿF#NGiven a function f such that f(z, n) = F(z) + O(p^n), where F is a power series with coefficients in the p-adic integers: F(z) = f_0 + f_1 z + f_2 z^2 + ... with Valuation(f_m) >= d*m, where d is non-negative, this function finds the coefficients f_m up to O(p^n), for m up to k. (Œ(¤(G#þÿX#ûÿûÿûÿûÿ$àŒ¼.H#
__sig__20Ü01I#àŒ ûÿÿÿÌ4O#àŒ8ûÿÿÿŒ7P#FindPowerSeriesForChabautyô9:Q#ÿÿÿÿÿÿÿÿ4R#./usr/local/Magma2.8/package/SchCrvHyp/reduce.m	hK	S#
./usr/local/Magma2.8/package/SchCrvHyp/reduce.m	hK	T$6/usr/local/Magma2.8/package/CrvMod/class_polynomials.m	`E	U$
$
__sig__0$,ûÿÿÿU#R,Given a degree d binary form F(x,z) with real coefficients, represented by d and pol = F(x,1) such that F is squarefree, returns its covariant in the upper half-plane. If Simple is set, the easier to compute Q_0-covariant is returned. Otherwise, the value of theta(pol) is returned as a second value.W¤WV#VVûÿûÿûÿ`LTP\Z#ÿÿÿÿÿÿÿÿX[#LGiven a degree d binary form F(x,z) with real coefficients, represented by d and pol = F(x,1) such that F is squarefree, returns a GL_2(Z)-equivalent form (in dehomogenized form) and the transformation matrix (an element of GL_2(Z)). If Simple is set, Q_0-reduction is used.Ðkèk\#kûÿûÿûÿ€xLl¼o^#Ltûÿÿÿds_#LGiven a degree d binary form F(x,z) with real coefficients, represented by d and pol = F(x,1) such that F is squarefree, returns a GL_2(Z)-equivalent form (in dehomogenized form) and the transformation matrix (an element of GL_2(Z)). If Simple is set, Q_0-reduction is used.Ì~`#kûÿûÿûÿˆL|`†a#
__sig__2‡ø‡ˆb#L„ûÿÿÿdŒ#
ReducedBasis,¬,	#ÿÿÿÿÿÿÿÿŒ
#(‚Computes a reduced basis (with respect to the height pairing) of the subgroup in J(Q)/torsion generated by the given points on J. Ä9Ü9#ÿûÿûÿûÿ<Œ4ŒŒ(Œ¼<#
__sig__1=ô=>#Œ0Œûÿÿÿ¤?#(‚Computes a reduced basis (with respect to the height pairing) of the subgroup in J(Q)/torsion generated by the given points on J. lF”F#ÿûÿûÿûÿìïDŒŒ8Œ°L#
__sig__2MNN#Œ@ŒûÿÿÿèO#./usr/local/Magma2.8/package/SchCrvHyp/height.m	¸S	#./usr/local/Magma2.8/package/SchCrvHyp/height.m	¸S	#
JacHypHeightV”V	$GReduce the coefficients of a genus 2 curve defined over the Rationals. W#
__sig__0X#L\ûÿÿÿ#¢kIf J is the Jacobian of a gneus 2 curve defined over the rationals, of the form y^2 = f(x) with integral coefficients, this computes a real number c such that h_K(P) le h(P) + c for all P in J(Q), where h_K is the naive logarithmic height got from the Kummer surface and h is the canonical height on J. Effort is some specification as to how much effort should be put into getting a good bound. The second value returned is a bound for mu_infinity. If Factor is set to true, the discriminant will be factored, and its prime divisors will be considered individually. This usually results in an improvement of the bound. œyäy#VVûÿûÿûÿ`ŒLŒTŒ¬~#ÿÿÿÿÿÿÿÿXŒ#7Computes the logarithmic height on the Kummer surface. ¸‡ø‡ˆ#
__sig__1‹@‹x‹q$
__sig__1\„r$ ‘Ä‘ûÿÿÿ˜s$
OvalDerivationŒ
t$ÿÿÿÿÿÿÿÿÀ‘u$A Baer subplane of PPlÀv$½½ÁÀûÿûÿà‘ ‘Ô‘w$
__sig__2Øðx$ ‘Ü‘ûÿÿÿy$
BaerSubplanez$ÿÿÿÿÿÿÿÿØ‘{$(/usr/local/Magma2.8/package/FldRe/real.mÐU	H`	|$
(/usr/local/Magma2.8/package/FldRe/real.mV	H`	}$WA value y and an error estimate dy, such that p(x) = y where p(xa[i]) = ya[i] for all i@è~$VÿVÿVVVûÿûÿûÿ’ð‘ô‘$$
__sig__0%(%@%€$ð‘ü‘ûÿÿÿ($\Approximation to integral of real function f from a to b, using Romberg's method of order 2k.,.‚$VVãÿVûÿûÿûÿûÿ’ð‘’ì5ƒ$
__sig__17d7Œ7„$ð‘’ûÿÿÿÜ9…$RombergQuadrature;,;†$ÿÿÿÿÿÿÿÿ’‡$ bApproximation to integral of real function f from a to b using Simpson's rule with n sub-intervals„A¼Aˆ$VVãÿVûÿûÿûÿûÿ(’ð‘’DF‰$
__sig__2HdH¤HŠ$ð‘$’ûÿÿÿtJ‹$SimpsonQuadraturehMˆMŒ$ÿÿÿÿÿÿÿÿ ’•$
__sig__4ture`Q$VApproximation to integral of f from a to b using trapezoidal rule with n sub-intervalsŒUŽ$VVãÿVûÿûÿûÿûÿ@’ð‘4’Z$
__sig__3[´[Ì[$ð‘<’ûÿÿÿÐ]‘$TrapezoidalQuadratureˆ`’$ÿÿÿÿÿÿÿÿ8’“$The Beta function B(z, w)@d°d”$VVVûÿûÿûÿûÿX’ð‘L’ˆg–$ð‘T’ûÿÿÿˆk—$
BetaFunctionl˜l˜$ÿÿÿÿÿÿÿÿP’0F)/usr/local/Magma2.8/package/RngLoc/misc.mð]	¨_	0JThe uniformizing element (an element with valuation 1) of the local ring L	ÿÿÿÿw0ûÿûÿûÿûÿ\Á$ÀÀÀ	0À Àûÿÿÿd}
04The generator of the maximal unramified subring of L€ 2
.m.ml™$=,/usr/local/Magma2.8/package/AlgGeom/specfile‡ˆ 0
__sig__0.8/package/0
	0ü
	|´	±#<ÑCompute the Igusa-Clebsch invariants of the curve y^2 + h*y - f = 0. The polynomial h must have degree at most 3, and the polynomial f must have degree at most 6. These will be all be zero in characteristic 2.Øð²#ûÿûÿûÿûÿØŽÐŽèÄŽ0´#èÌŽûÿÿÿ\µ#4±Compute the Igusa J-invariants of a genus-2 curve over a field. These will be all be zero in characteristic 2. If Quick, the base field must not be of characteristic 2, 3, or 5.$¶#gûÿûÿûÿûÿàŽèÔŽ·#
	__sig__12lŒ¸#èÜŽûÿÿÿè¹#<ÕCompute the 10 absolute invariants as on p. 325 by Mestre: 	J_2^5/J_10, J_2^3*J4/J_10, J_2^2*J_6/J_10, J_2*J_8/J_10, J_4*J_6/J_10, 	J_4*J_8^2/J_10^2, J_6^2*J_8/J_10^2, J_6^5/J_10^3, J_6*J_8^3/J_10^3, 	J_8^5/J_10^4¼*º#gûÿûÿûÿûÿðŽèäŽl0¼#èìŽûÿÿÿ¼3¥$„’’ûÿÿÿeom/misc.mš$*/usr/local/Magma2.8/package/AlgGeom/misc.m e	€q	›$*/usr/local/Magma2.8/package/AlgGeom/misc.mhe	€q	œ$]Returns a scheme defined by the equations of X interpreted over R using the default coercion.H$ýÿßßûÿûÿûÿûÿ|’l’p’ðž$
__sig__0ÈàŸ$l’x’ûÿÿÿ@T*
__sig__0.8/package/`*Spanys.m	H&
__sig__0ge/AlgGeom/ $3/usr/local/Magma2.8/package/AlgGeom/Aux/polyrings.mhg	€q	¡$*3/usr/local/Magma2.8/package/AlgGeom/Aux/polyrings.m¸g	€q	¢$DThe polynomial comprising terms of f whose total degree is that of f#Ä#£$KKûÿûÿûÿûÿ”’„’ˆ’H(¦$

DegreeForm *À*§$ÿÿÿÿÿÿÿÿŒ’¨$The constant term of f¸0©$KKûÿûÿûÿûÿ¬’„’ ’7ª$
__sig__19h9 9«$„’¨’ûÿÿÿ;¬$
ConstantTerm<0<­$ÿÿÿÿÿÿÿÿ¤’®$CThe linear term of f with respect to the grading on its parent ringˆAÀA¯$KKûÿûÿûÿûÿÄ’„’¸’HF°$
__sig__2HhH¨H±$„’À’ûÿÿÿxJ²$

LinearTerm4MTM³$ÿÿÿÿÿÿÿÿ¼’´$JThe affine linear term of f with respect to the grading on its parent ringS Sµ$KKûÿûÿûÿûÿÜ’„’Ð’xW¶$
__sig__3XèX0Y·$„’Ø’ûÿÿÿ [¸$AffineLinearTerm]L]t]¹$ÿÿÿÿÿÿÿÿÔ’º$QThe terms of f of weight at most n with respect to the grading on its parent ringDd´d»$KKûÿûÿûÿûÿô’„’è’Œg¼$
__sig__4jk<k½$„’ð’ûÿÿÿœl¾$
GradedSubTerm¬m¿$ÿÿÿÿÿÿÿÿì’À$IThe terms of f of weight n with respect to the grading on its parent ringðuvÁ$KKûÿûÿûÿûÿ“„’“|yÂ$
__sig__5zÈzàzÃ$„’“ûÿÿÿ¨}Ä$

GradedTerm 8Å$ÿÿÿÿÿÿÿÿ“Æ$PA sequence containing all the derivatives of order m of all the polynomials in f‹D‹|‹È$
__sig__6Ž(ŽPŽ½#AbsoluteInvariantsh77¾#ÿÿÿÿÿÿÿÿèŽÍ#
	JacHypCntsch€;¿#7Convert Clebsch invariants to Igusa-Clebsch invariants Ð=ø= >À#ûÿûÿûÿûÿèüŽBÁ#
	__sig__14PDÈDÂ#èûÿÿÿFÃ#ClebschToIgusaClebsch¨HÄ#ÿÿÿÿÿÿÿÿÅ#7Convert Igusa-Clebsch invariants to Clebsch invariants \O„OœOÆ#ûÿûÿûÿûÿ è SÇ#
	__sig__15XT€TÈ#èûÿÿÿ˜VÉ#IgusaClebschToClebschÐXÊ#ÿÿÿÿÿÿÿÿË#1/usr/local/Magma2.8/package/SchCrvHyp/jac_count.m(u	Xy	Ì#1/usr/local/Magma2.8/package/SchCrvHyp/jac_count.mxu	Xy	Î#9The order of the Jacobian J defined over a finite field. \eteÏ#ûÿûÿûÿûÿÌØD08œjÐ#
__sig__0kllÑ#0@ûÿÿÿ|mÒ#(„The Euler factor of a Jacobian over a finite field computed modulo p, using the Hasse-Witt matrix. Warning: cost in M(g.p).log(g.p) u¨uÓ#ûÿûÿûÿûÿT0HøxÔ#
__sig__1z@zXzÕ#0PûÿÿÿÄ|Ö#EulerFactorModCharÐ~×#ÿÿÿÿÿÿÿÿLØ#4/usr/local/Magma2.8/package/SchCrvHyp/Genus2/G2.spec‡ˆç#7/usr/local/Magma2.8/package/SchCrvHyp/Genus2/cardmod3.mXy	Hi
	ä»ç¡#
JInvariantsôvw¢#ÿÿÿÿÿÿÿÿxŽ£#6¼Compute the Igusa J-invariants of a polynomial of degree at most 6. The integer 2 must be a unit of the coefficient ring. If Quick, the base field must not be of characteristic 2, 3, or 5.…È…¤#ûÿûÿûÿûÿ Ž˜ŽèŒŽhŒ|%$pGiven a nonsingular point or place p of a curve, returns an element of the function field of valuation one at p.
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ì
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	__sig__16h%Ô”ø•ûÿÿÿ€%%The function field place underlying P`%þAûÿûÿûÿûÿ–Ô”–\ƒ%Ô”–ûÿÿÿ|„%FunctionFieldPlaceœ¼…%ÿÿÿÿÿÿÿÿ–†%"The curve place corresponding to Pt”‡%Aþûÿûÿûÿûÿ$–Ô”–€ˆ%
	__sig__18<‰%Ô” –ûÿÿÿ° Š%

CurvePlace""‹%ÿÿÿÿÿÿÿÿ–”%
	__sig__20or&`&Œ%'The function field divisor underlying D)\*t*%?ûÿûÿûÿûÿ<–Ô”0–ü/Ž%
	__sig__19\1t1%Ô”8–ûÿÿÿ,6%FunctionFieldDivisor8l8‘%ÿÿÿÿÿÿÿÿ4–’%$The curve divisor corresponding to D<Œ<“%?ûÿûÿûÿûÿT–Ô”H–Ô?•%Ô”P–ûÿÿÿB–%
CurveDivisorDÄD—%ÿÿÿÿÿÿÿÿL–˜% _The factorisation sequence of D; each term is of the form <place of a curve, integer valuation>PMhMˆM™%ûÿûÿûÿûÿì¢l–Ô”`–`Qš%
	__sig__21ŒR¤R›%Ô”h–ûÿÿÿ<Tœ%&yTrue and a place of degree m on the curve C defined over a finite field if and only if there exists such; false otherwise<[T[%éþûÿûÿûÿ|–Ô”p–H`ž%
	__sig__22@aXaŸ%Ô”x–ûÿÿÿxb %GA sequence of places of degree m on a curve defined over a finite fieldf(f@f¡%éûÿûÿûÿûÿè¶Œ–Ô”€–èk¢%
	__sig__23ðlm£%Ô”ˆ–ûÿÿÿ|n¤%GTrue and a random place of degree m of C if one exists, otherwise falseÌtätu¥%éþûÿûÿûÿœ–Ô”–x¦%
	__sig__24`yxy§%Ô”˜–ûÿÿÿ{¨%
0}d}©%þûÿûÿûÿûÿ´–¬–Ô” –„‚ª%
	__sig__25(†`†«%Ô”¨–ûÿÿÿ ˆ¬%;True iff the degree 1 place P is a Weierstrass place (of D)ôŽ­%þûÿûÿûÿûÿ¼–Ô”°–˜’Ç$ûÿûÿûÿûÿ4“$“„’“ÔÉ$„’ “ûÿÿÿ`Ê$
AllDerivativesPË$ÿÿÿÿÿÿÿÿ“Ì$9A sequence containing all the derivatives of order m of f¤¼Í$Kûÿûÿûÿûÿ<“„’0“lÎ$
__sig__7¤¼Ï$„’8“ûÿÿÿ$Ð$]The #I-th derivative obtained by differentiating with respect to the I[i]-th variable in turn€Ñ$ÿKKûÿûÿûÿûÿL“„’@“,!Ò$
__sig__8"€"˜"Ó$„’H“ûÿÿÿh$Õ$TThe sequence S reversed and translated so that the last term is in the n-th positionactors Ô$ûÿûÿûÿûÿX“h¾T“Ö$h¾\“ûÿÿÿ1/%Internal functionÿÿŒ6Ø$
RemoveFactor8œ8Ù$ÿÿÿÿÿÿÿÿˆ¾Ú$ cThe common denominator of the rational functions in F and the normalised sequence after its removal¼>Ü>Û$ÿâÿKûÿûÿûÿt“„’h“¸CÜ$
	__sig__10\E„EÝ$„’p“ûÿÿÿ$GÞ$CommonDenominatorœI´Iß$ÿÿÿÿÿÿÿÿl“à$$vThe homogenisation of f in degree d with respect to z if this is possible with the grading on their common parent ringtRá$KKKûÿûÿûÿûÿœ“Œ“„’€“\Vâ$
	__sig__11ìWXã$„’ˆ“ûÿÿÿœZê$*The lowest degree among the monomials of fH]p]ë$Kûÿûÿûÿûÿ´“„’¨“¸aí$„’°“ûÿÿÿØbî$$tThe sequence of least degrees of f with respect to each different weighting given by the columns of the 2x2 matrix Mgøgï$ïÿKûÿûÿûÿÄ“„’¸“xmð$
	__sig__14änŒoñ$„’À“ûÿÿÿsò$
MinimumDegreesÌtó$ÿÿÿÿÿÿÿÿ¼“ô$(The polynomial term of least degree of fxÌxôxõ$KKûÿûÿûÿûÿÜ“„’Ð“¤}ö$
	__sig__154÷$„’Ø“ûÿÿÿ„ø$MinimumDegreeTerm(†`†ù$ÿÿÿÿÿÿÿÿÔ“ú$$The polynomial term of degree d in fŽ¼Žü$
	__sig__164\7$CanonicalModularEquation ¬^;8$ÿÿÿÿÿÿÿÿÐ9$*The classical modular equation of level p. :$Kûÿûÿûÿûÿð˜ä;$
__sig__3 ’`;<$˜ìûÿÿÿ=$ClassicalModularEquation—	>$ÿÿÿÿÿÿÿÿè?$6/usr/local/Magma2.8/package/CrvMod/modular_isogenies.m	X˜	@$6/usr/local/Magma2.8/package/CrvMod/modular_isogenies.m	X˜	A$ eReturns the sequence of points on X specifying isogenies of the elliptic curve E over its base field.ŒB$&%ûÿûÿûÿûÿ‘‘‘|	D$‘‘ûÿÿÿdmod3.mmÙ#:/usr/local/Magma2.8/package/SchCrvHyp/Genus2/weilpairing.mX˜	¸¢	Ú#:/usr/local/Magma2.8/package/SchCrvHyp/Genus2/weilpairing.m°˜	¸¢	Û#5The Weil Pairing e_m(P, Q) where P and Q are in J[m] ü Ü#þÿûÿûÿûÿûÿxhl¸$Ý#
__sig__0&ˆ&Ð&Þ#htûÿÿÿ¤(ß#4/usr/local/Magma2.8/package/SchCrvHyp/Genus2/count.mš	¸¢	à#8/usr/local/Magma2.8/package/SchCrvHyp/Genus2/divpoljac.m¢	hš	¸¢	å#5/usr/local/Magma2.8/package/SchCrvHyp/Genus2/schoof.m	›	Hy	ä»çâ#
	PsiDivPolô9:û$KKûÿûÿûÿûÿô“„’è“Hý$„’ð“ûÿÿÿ$þ$
HomogeneousTerm˜ÿ$ÿÿÿÿÿÿÿÿì“%
__sig__0nerated%:The sequence of coefficients of monomials in f of degree dØð%Kûÿûÿûÿûÿ””„’”„%
	__sig__17„œ%„’”ûÿÿÿ<%CoefficientsOfDegree”%ÿÿÿÿÿÿÿÿ”% `The sequence of coefficients of monomials in f of bidegree d,e with respect to the gradings on F  ° %Ksûÿûÿûÿûÿ$”„’”h$%
	__sig__18H&`&	%„’ ”ûÿÿÿt(%HTrue iff the current set of generators of I are individually homogeneousŸ	”“	
%Lûÿûÿûÿûÿ0”„’,”¼ç %Wedge,7%PolynomialOfDegreed9œ9%ÿÿÿÿÿÿÿÿl”%<The polynomial on F of bidegree d,e with coefficients from c=>%ÿþÿsKûÿûÿûÿûÿL”„’@”B%
	__sig__20LDÄD%„’H”ûÿÿÿF%
	__sig__22eratedH%0/usr/local/Magma2.8/package/AlgGeom/Aux/matrix.mndiÐ¡	ø¢	%0/usr/local/Magma2.8/package/AlgGeom/Aux/matrix.m ¢	ø¢	%+A sequence containing the r x r minors of M	__si%P”x”ûÿÿÿûÿÿÿ%IsHomogeneouslyGeneratedV|V”V%ÿÿÿÿÿÿÿÿ(”%8The polynomial of R of degree d with coefficients from c]H]p]%ÿþÿLKûÿûÿûÿûÿD”t”„’h”¸a%„’p”ûÿÿÿØb%
	__sig__23øde%„’4”ûÿÿÿ@f%ïÿûÿûÿûÿûÿ|”P”\”èk!%ÿÿÿÿÿÿÿÿX”"%0A sequence containing all the r by r minors of MrÜrs#%ïÿûÿûÿûÿûÿ””P”ˆ”€v$%
__sig__1w¼wÔw%%P””ûÿÿÿxy'%ÿÿÿÿÿÿÿÿŒ”(%ZTrue if and only if all entries of M are coercible into S; in that case, also return 'S!M'‚„‚)%ïÿ÷ÿïÿûÿûÿûÿ¬”P” ” ‰*%
__sig__2(@+%P”¨”ûÿÿÿÐL%
¬Ä+$
ModularEquation¤,$ÿÿÿÿÿÿÿÿ C$
__sig__0quation-$,The Atkin modular equation of prime level p.ð.$KûÿûÿûÿûÿÀ˜´0/$
__sig__1,0$˜¼ûÿÿÿL1$AtkinModularEquation„2$ÿÿÿÿÿÿÿÿ¸3$0The canonical modular equation of prime level p.ÄÜ4$KûÿûÿûÿûÿØ˜Ìè5$
__sig__2°6$˜Ôûÿÿÿü E$
ModuliPoints"h"F$ÿÿÿÿÿÿÿÿ‘G$CThe subgroup scheme of E defined by the point P on a modular curve.¼()H$í&ûÿûÿûÿûÿ(‘‘‘l/I$
__sig__111D1J$‘$‘ûÿÿÿì5K$TReturns the isogeny E -> F corresponding to the point P on the modular curve X_0(N).:œ:L$í&ûÿûÿûÿûÿ8‘‘,‘|=M$
__sig__2>¼>Ü>N$‘4‘ûÿÿÿ4@O$3/usr/local/Magma2.8/package/CrvMod/velu_isogenies.mØ­	´	P$,/usr/local/Magma2.8/package/CrvMod/lercier.m®	´	Q$+/usr/local/Magma2.8/package/CrvMod/elkies.m®	´	R$
ModularIsogenyˆMN<§§ûÿûÿûÿûÿ@ñ,ñ4ñ‘<\øs/magma/ÿÿÿÿ“ÿÿÿÿmisc.mm.8/q<
__sig__0mls.mm04
__sig__1k$__sig__0iallnomial”XV$ûÿûÿûÿûÿ`‘P‘T‘ ]W$
__sig__0_P_h_X$P‘\‘ûÿÿÿ(aY$WeberToHilbertClassPolynomialxbZ$ÿÿÿÿÿÿÿÿX‘[$+Returns the Hilbert class polynomial for D.èfg\$ûÿûÿûÿûÿx‘P‘l‘Èl]$
__sig__1mÀmØm^$P‘t‘ûÿÿÿ„p_$HilbertClassPolynomial4t`$ÿÿÿÿÿÿÿÿp‘a$\The class polynomial of binary quadratic forms of discrimant D, in terms of Weber functions.yzb$ûÿûÿûÿûÿ‘P‘„‘c$
__sig__2ô€d$P‘Œ‘ûÿÿÿÄ…e$WeberClassPolynomial‡¸‡f$ÿÿÿÿÿÿÿÿˆ‘¤$
__sig__0.8/package/ï#ÿÿÿÿÿÿÿÿ¬ð#$vTrue if the genus 2 curve with the given invariants have an automorphism different from the hyperelliptic involution. ¼Añ#ûÿûÿûÿûÿÜÌ¤ÀDFò#
__sig__1HdH¤Hó#¤ÈûÿÿÿtJô#HasManyAutomorphismsMˆMõ#ÿÿÿÿÿÿÿÿÄö#[True if the genus 2 curve has an automorphism different from the hyperelliptic involution. ìST÷#gûÿûÿûÿûÿä¤Ø”Xø#
__sig__2ZtZœZù#¤àûÿÿÿP\ú# bCompute the conic L and the cubic M for the IgusaClebsch invariants; tool for Mestre's algorithm.  a¸aû#ûÿûÿûÿûÿô¤èpeý#¤ðûÿÿÿ fþ#
ConicLAndCubicMøgÿ#ÿÿÿÿÿÿÿÿì$8Find an affine point (x,y,1) on the projective conic L. o¤o¼o$Kÿÿûÿûÿûÿûÿ¤tu$
__sig__4v\v€v$¤ûÿÿÿx$FindPointOnConicyœyäy$ÿÿÿÿÿÿÿÿ$//usr/local/Magma2.8/package/SchCrvHyp/wam_red.mø»	$//usr/local/Magma2.8/package/SchCrvHyp/wam_red.mø»	$
	CrvHypRed‡¸‡å<Üóûÿÿÿym/e64
__sig__2°Ð>4
H‘”ÃL&8The local intersection number of the curves C and D at pdœM&íééûÿûÿûÿûÿD™<™™0™œN&
__sig__1|”O&™8™ûÿÿÿÄP&The intersection number (C,D)ÈQ&ééûÿûÿûÿûÿL™™@™PR&
__sig__28XS&™H™ûÿÿÿ T&"The intersection points of C and D ÀU&ééûÿûÿûÿûÿ”™\™™P™V&
__sig__3ÌìW&™X™ûÿÿÿÜX&IntersectionPoints!0!Y&ÿÿÿÿÿÿÿÿT™[&CThe number of geometric punctures of the affine curve C at infinityÔ˜	Z&éûÿûÿûÿûÿp™™l™,¼ç]&
__sig__7À	Ü˜	_&3/usr/local/Magma2.8/package/AlgGeom/curve/tangent.mpÀ	°Á	^&3/usr/local/Magma2.8/package/AlgGeom/curve/tangent.mÀÀ	°Á	`&íééûÿûÿûÿûÿ´™¤™x™„™<h&
__sig__0™€™ûÿÿÿb&NumberOfPuncturesÀ?Ø?c&ÿÿÿÿÿÿÿÿh™d&1The intersection points of C and D defined over K0FHFe&ýÿééûÿûÿûÿûÿœ™™™ôKf&
__sig__6M¤MÄMg&™˜™ûÿÿÿœO\&™t™ûÿÿÿdQi&x™ ™ûÿÿÿðRr&
__sig__2T(T@Ts&x™È™ûÿÿÿ`Vt&
TangentConeðWXu&ÿÿÿÿÿÿÿÿÄ™v&ITrue iff C and D have a common tangent cone component at rational point p4`L`w&íééûÿûÿûÿûÿä™x™Ø™Übx&
__sig__3düdey&x™à™ûÿÿÿDfz&
	IsTangentTgŒg{&ÿÿÿÿÿÿÿÿÜ™|&1/usr/local/Magma2.8/package/AlgGeom/curve/param.màÅ	hÉ	}&1/usr/local/Magma2.8/package/AlgGeom/curve/param.m0Æ	hÉ	&IsLineu~&YParametrization for the algebraic function field F defined by the divisor D of degree onex¨x&?àáSSûÿûÿššô™ø™h}€&
__sig__0~Ð~&ô™šûÿÿÿÈ€‚&
ParametrizationÈ…ƒ&ÿÿÿÿÿÿÿÿü™„&
D‹|‹…&íé0ûÿûÿûÿûÿ$ššô™šÈ†&
__sig__1‘è‘’‡&ô™šûÿÿÿ|“Š&
__sig__2”œ”´”¨&
Genus—®%
	__sig__26$L¯%Ô”¸–ûÿÿÿ(±%þûÿûÿûÿûÿÔ–Ì–Ô”À–$²%
	__sig__27d³%Ô”È–ûÿÿÿ´%Degree of place or divisorH`µ%ûÿûÿûÿûÿT£Ü–Ô”Ð–¶%
	__sig__28(P·%Ô”Ø–ûÿÿÿÀ¸%
x˜¹%íûÿûÿûÿûÿô–ì–Ô”à–„º%
	__sig__29@»%Ô”è–ûÿÿÿ´ ¼%AThe coefficient in D of the place p or of the nonsingular place p$$<$½%þûÿûÿûÿûÿ™ü–Ô”ð–¨(¾%
	__sig__30+¸+¿%Ô”ø–ûÿÿÿ.Ò%
ivisorÀ%.Two effective divisors A,B such that D = A - BÐ4Á%ûÿûÿûÿ—Ô”—:Â%
	__sig__31;0;Ã%Ô”—ûÿÿÿ<Ä%SignDecompositionø= >Å%ÿÿÿÿÿÿÿÿ—Æ%&wThe principal divisor corresponding to the intersection with the curve L minus a contribution from the line at infinity˜FG(GÇ%éÿûÿûÿûÿûÿ,—$—Ô”—MÈ%
	__sig__32DN\NÉ%Ô” —ûÿÿÿ\PÊ%&~The principal divisor corresponding to the line through points p,q of the curve minus a contribution from the line at infinityÈUË%ííÿûÿûÿûÿûÿÀ˜4—Ô”(— ZÌ%
	__sig__33è[\Í%Ô”0—ûÿÿÿD^Î%A canonical divisor of C`Ô`ì`Ï%éûÿûÿûÿûÿD—Ô”8—´dÐ%
	__sig__34œe´eÑ%Ô”@—ûÿÿÿÔfÚ%

aryDivisorh,hÓ%íûÿûÿûÿûÿ\—T—Ô”H—¬mÔ%
	__sig__35¨oÀoÕ%Ô”P—ûÿÿÿhsÖ%"mThe divisor obtained from D by removing all components of the nonsingular point or place p have been removed.Øw×%þûÿûÿûÿûÿd—Ô”X—T{Ø%
	__sig__36à}H~Ù%Ô”`—ûÿÿÿ°Û%íûÿûÿûÿûÿt—Ô”h—‡Ü%
	__sig__37‰$‰Ý%Ô”p—ûÿÿÿŽ&
cialityÞ%]A vector space and the isomorphism from this space to the Riemann-Roch space of the divisor D,“à%
	__sig__38d”„”Q%þûÿûÿûÿûÿä–L•Ô”@•hR%
__sig__6´ÔS%Ô”H•ûÿÿÿôT%7The divisor whose factorisation sequence of tuples is Sx ÈU%ÿÿûÿûÿûÿûÿd•\•Ô”P•PV%
__sig__78XW%Ô”X•ûÿÿÿ X%^The principal divisor on the curve C of the divisor group D defined by the rational function fHY%þÿÿûÿûÿûÿûÿÌ•l•Ô”`•Z%
__sig__8ô[%Ô”h•ûÿÿÿœ!\%^The principal divisor on the curve C of the divisor group D defined by the rational function fd&]%þÿÿûÿûÿûÿûÿ|•Ô”p•,^%
__sig__9-è-._%Ô”x•ûÿÿÿ¸0`%?A sequence containing the zeros on C of the rational function f6è607a%þÿéûÿûÿûÿûÿŒ•Ô”€•0;b%
	__sig__10H<`<c%Ô”ˆ•ûÿÿÿÐ=d%?A sequence containing the poles on C of the rational function f0AHAhAe%þÿéûÿûÿûÿûÿœ•Ô”•Ff%
	__sig__11H(Hg%Ô”˜•ûÿÿÿJh%<A sequence of rational functions which determine the place pNœNi%þûÿûÿûÿûÿ¬•Ô” •xRj%
	__sig__12hSSk%Ô”¨•ûÿÿÿÈUl%RationalFunctionsW¨Wm%ÿÿÿÿÿÿÿÿ¤•n%XThe place of a curve C determined by the sequence of elements of the function field of C`t`Œ`o%þûÿûÿûÿûÿt–Ä•Ô”¸•|cp%
	__sig__13,eDeq%Ô”À•ûÿÿÿtfr%FþThe divisor of a curve C determined by the factorization sequence Q of elemnts of the form <Q,n> where Q is a sequence of elements of the function field of C determining a place and the integer n is the multiplicity of that place in the resulting divisorÐts%ûÿûÿûÿûÿ—Ô•Ô”È•Øwt%
	__sig__14(yLyu%Ô”Ð•ûÿÿÿàz‚%
	__sig__17terh}w%íáûÿûÿûÿûÿô•ä•Ô”Ø•ˆ‚x%
	__sig__15,†d†y%Ô”à•ûÿÿÿ¤ˆz%UniformizingParameter{%ÿÿÿÿÿÿÿÿÜ•}%þáûÿûÿûÿûÿü•Ô”ð•,“ä$
HomogenisationLyå$ÿÿÿÿÿÿÿÿ„“æ$"jThe homogenisation of f with respect to z if this is possible with the grading on their common parent ringÀ‚¬„ç$KKKûÿûÿûÿûÿ¤“„’˜“,‹è$
	__sig__12hŽé$„’ “ûÿÿÿì$
	__sig__13‘h‘ß%tûÿûÿûÿ„—Ô”x—Há%Ô”€—ûÿÿÿ$â%&A basis of the Riemann-Roch space L(D)0ã%ûÿûÿûÿûÿ\Æ”—Ô”ˆ—Dä%
	__sig__39H`å%Ô”—ûÿÿÿŒæ%The dimension of L(K_C - D)ÄÜç%ûÿûÿûÿûÿ¤—Ô”˜—è%
	__sig__40$é%Ô” —ûÿÿÿ|ê%True iff D is the zero divisorë%ûÿûÿûÿûÿ´—Ô”¨—° ì%
	__sig__41""í%Ô”°—ûÿÿÿ$ï%True iff D is specialep%î%ûÿûÿûÿûÿÀ—Ô”¼—Ô”ñ%
	__sig__44sig__42ð%Ô”Ä—ûÿÿÿ´0ò%True iff D is effectiveô34Ì4ó%ûÿûÿûÿûÿÔ—Ô”È—:ô%
	__sig__43;,;õ%Ô”Ð—ûÿÿÿŒ<ö%HTrue iff D is a principal divisor, in which case also return a generator?¼?Ô?÷%áûÿûÿûÿä—Ô”Ø—DEø%
	__sig__45,FDFù%Ô”à—ûÿÿÿôH&
	__sig__47enttJú%%True iff D - E is a principal divisorNû%ûÿûÿûÿûÿô—Ô”è—àQü%
	__sig__46SSý%Ô”ð—ûÿÿÿ\U	&Ô” ˜ûÿÿÿW
&
LX”X&êþ0ûÿûÿûÿûÿ<˜4˜Ô”(˜ ]&
	__sig__49P_h_&Ô”0˜ûÿÿÿ(a&OThe map to the projective space P determined by the complete linear system of Dàdøde&ê0ûÿûÿûÿûÿt˜D˜Ô”8˜øg&
	__sig__50 k¸k&Ô”@˜ûÿÿÿma&
nctures&
¤o¼o&éûÿûÿûÿûÿd˜T˜Ô”H˜tu&
	__sig__51\v€v&Ô”P˜ûÿÿÿx&
CanonicalMapyxy&ÿÿÿÿÿÿÿÿL˜& _The map determined by the canonical linear system of C to the projective space P if not omitted„‚¼‚¨„&êéûÿûÿûÿûÿl˜Ô”`˜(‹&
	__sig__52dü&Ô”h˜ûÿÿÿ&JThe map determined by the divisor n*p, where p is a nonsingular point on C“(“&	__sig__53`”€”˜”°”›'ßûÿûÿûÿûÿ0±tž<žhžLœ'
__sig__2
Ø
ð
'<žpžûÿÿÿœž'True iff X is reducibleÄÜôŸ'ßûÿûÿûÿûÿ„ž<žxž4 '
__sig__30¡'<ž€žûÿÿÿP¢'
IsReducible8X£'ÿÿÿÿÿÿÿÿ|ž¤':A sequence containing the prime components of the scheme XÈà¥'ßûÿûÿûÿûÿœž<žžì¦'
__sig__4´§'<ž˜žûÿÿÿ!¨'
PrimeComponentsl"©'ÿÿÿÿÿÿÿÿ”žª'<A sequence containing the primary components of the scheme X(¨(«'ßûÿûÿûÿûÿ´ž<ž¨žÀ.¬'
__sig__50à01­'<ž°žûÿÿÿÐ4®'PrimaryComponentsh77¯'ÿÿÿÿÿÿÿÿ¬žº'
__sig__0nents€;°'BA sequence containing the minimal prime components of the scheme XH>p>±'ßûÿûÿûÿûÿÌž<žÀž¨B²'
__sig__6DEE³'<žÈžûÿÿÿHF´'MinimalPrimeComponentsøHµ'ÿÿÿÿÿÿÿÿÄž¶'3/usr/local/Magma2.8/package/AlgGeom/scheme/subset.mhù	H	·'3/usr/local/Magma2.8/package/AlgGeom/scheme/subset.m¸ù	H	¹'ßßûÿûÿûÿûÿôžìžÜžàž`V»'Üžèžûÿÿÿ˜X¼'
ØYZ½'ßûÿûÿûÿûÿŸüžÜžðž<_¾'
__sig__1`¤`¼`¿'ÜžøžûÿÿÿìaÀ'GTrue iff the scheme X is contained in Y or all the points in S lie in Xteœe´eÁ'ßûÿûÿûÿûÿàÇŸÜžŸ<kÂ'
__sig__2l4lLlÃ'ÜžŸûÿÿÿ¬mÄ'8/usr/local/Magma2.8/package/AlgGeom/scheme/singularity.ms¨ü	H	Å'(8/usr/local/Magma2.8/package/AlgGeom/scheme/singularity.mvý	H	Æ'*The codimension of X in its ambient space.x¨xÇ'ßûÿûÿûÿûÿ@­$ŸŸŸh}È'
__sig__0~Ð~É'Ÿ ŸûÿÿÿÈ€Ê'
Codimension…È…Ë'ÿÿÿÿÿÿÿÿŸà'
__sig__4hemehŒÌ'IThe subscheme of X of points at which the jacobian matrix of X drops rankàøÍ'ßßûÿûÿûÿûÿ<ŸŸ0Ÿ<”Î'
__sig__1–,–D–Ï'Ÿ8ŸûÿÿÿšÐ'JacobianSubrankSchemeÔ›ï&ÿÿÿÿÿÿÿÿ¬›ð&<ÖA curve W in Weierstrass form isomorphic to the nonsingular genus 1 curve C; the second return value is a map of the ambient space restricting to an isomorphism C -> W taking the point p to the the flex at infinitydñ&íéé0ûÿûÿûÿÌ›Œ›À›àò&
__sig__2èó&Œ›È›ûÿÿÿ€ô&WeierstrassFormNonFlexðõ&ÿÿÿÿÿÿÿÿÄ›ö&:/usr/local/Magma2.8/package/AlgGeom/curve/special_curves.m¨	¸	÷&:/usr/local/Magma2.8/package/AlgGeom/curve/special_curves.m	¸	ø&-The Klein Quartic in the projective plane P2.<$ù&êéûÿûÿûÿûÿì›Ü›à›¨(û&Ü›è›ûÿÿÿ,'0/usr/local/Magma2.8/package/AlgGeom/scheme/sch.m08	¸	'The empty subscheme of X3à3ø3'ßßûÿûÿûÿûÿ$œœœà9'
__sig__0:è:;	'œ œûÿÿÿ`<
'2True iff the ambient space of X is an affine spaceÀ>à>'ßûÿûÿûÿûÿ4œœ(œ¼C'
__sig__1E`EˆE'œ0œûÿÿÿ(G'
IsAffineH0IxI'ÿÿÿÿÿÿÿÿ,œ'5True iff the ambient space of X is a projective spaceìO'ßûÿûÿûÿûÿLœœ@œPS'
__sig__2T˜T`U'œHœûÿÿÿW"'
__sig__5paceÐX'NTrue iff the ambient space of X is some ordinary (unweighted) projective spacet]'ßûÿûÿûÿûÿ\œœPœ¼a'
__sig__3b”b¬b'œXœûÿÿÿe'IsOrdinaryProjectivefDf'ÿÿÿÿÿÿÿÿTœ'.True iff X and Y lie in the same ambient spaceÌl'ßßûÿûÿûÿûÿtœœhœÈr'
__sig__4t€t t'œpœûÿÿÿØu'IdenticalAmbientSpacepw'ÿÿÿÿÿÿÿÿlœ '#A Groebner basis for the ideal of X|{¬{!'ßûÿûÿûÿûÿŒœœ€œÈ€#'œˆœûÿÿÿ†$'"A minimal basis for the ideal of X¼ˆÔˆ%'ßûÿûÿûÿûÿœœœœ&'
__sig__6‘h‘''œ˜œûÿÿÿÌ’('QTrue iff X is defined by linear equations, possibly after taking a Groebner basis\˜,™*'__sig__7šœš´šÜšôš½&ÿÿÿÿÿÿÿÿäš¾&!True iff p is a double point of CÀØ¿&íûÿûÿûÿûÿ›lšøšÄÁ&lš›ûÿÿÿxÃ&íéûÿûÿûÿûÿ$››lš› Ä&
__sig__7ø Å&lš›ûÿÿÿpÆ&IsNode€Ç&ÿÿÿÿÿÿÿÿ›È&+True iff p is an ordinary double point of C8XÉ&íûÿûÿûÿûÿ,›lš ›Ê&
__sig__8!!0!Ë&lš(›ûÿÿÿL#Ì&
$$<$Í&íéûÿûÿûÿûÿL›<›lš0›¨(Î&
__sig__9*+¸+Ï&lš8›ûÿÿÿ.Ð&IsCusp0Ñ&ÿÿÿÿÿÿÿÿ4›Ò&-True iff p is a nonordinary double point of Cp8Ó&íûÿûÿûÿûÿ,ëT›lšH›<Ô&
	__sig__10Ø<ð<Õ&lšP›ûÿÿÿ¨>Ö&6/usr/local/Magma2.8/package/AlgGeom/curve/blowup_crv.m		×&6/usr/local/Magma2.8/package/AlgGeom/curve/blowup_crv.m		Ø&8The two natural patches of the blowup of C at the originI I¸IÙ&éééûÿûÿûÿ|›l›\›`›œNÚ&
__sig__0OP\PÛ&\›h›ûÿÿÿäQÜ&BlowupðRÝ&ÿÿÿÿÿÿÿÿd›Þ&?The birational transform of C under the toric blowup given by M0YHYYß&ïÿééûÿûÿ„›\›x›D^à&
__sig__1`t`Œ`á&\›€›ûÿÿÿ¼aâ&7/usr/local/Magma2.8/package/AlgGeom/curve/weierstrass.m	ã&7/usr/local/Magma2.8/package/AlgGeom/curve/weierstrass.m	~'Flexessä&$vAn elliptic curve E in Weierstrass form isomorphic to the (closure of the) nonsingular cubic curve C with a map E -> Coå&íé&ûÿûÿûÿœ›Œ››Huæ&
__sig__0v0vHvç&Œ›˜›ûÿÿÿØwè&
WeierstrassFormLyé&ÿÿÿÿÿÿÿÿ”›ê&@ãA curve W in Weierstrass form isomorphic to the nonsingular genus 1 curve C. The second return value is a translation of the ambient space restricting to an isomorphism C -> W taking the rational point p to the flex at infinity$ŒhŒë&íé&ûÿûÿûÿ´›Œ›¨›øì&
__sig__1’0’H’í&Œ›°›ûÿÿÿÌ“î&WeierstrassFormFlexì•–ú&__sig__0nFlexŒ™¬™Ô™j&
TangentLineÀwØwk&ÿÿÿÿÿÿÿÿ€™l& `Tangent line to C at the nonsingular rational point p embedded as a curve in the same plane as C°m&íéûÿûÿûÿûÿ¼™x™°™‡n&
__sig__1ˆ‰$‰o&x™¸™ûÿÿÿŽp&XThe tangent cone to C at the rational point p embedded as a curve in the same plane as C’0’H’q&íééûÿûÿûÿûÿì¡Ì™x™À™D–&íéûÿûÿûÿûÿ|˜Ô”p˜	&Ô”x˜ûÿÿÿ¨ &*‰The divisor class group of the curve C defined over a finite field together with the map from the divisor group of C onto the class group È!&é“ûÿûÿûÿŒ˜Ô”€˜P"&
	__sig__548X#&Ô”ˆ˜ûÿÿÿ $&*A basis of the space of differentials of DØð%&ûÿûÿûÿûÿœ˜Ô”˜à&&
	__sig__55X'&Ô”˜˜ûÿÿÿ(&]A vector space and the isomorphism from this space to the differential space of the divisor D<$)&çÿûÿûÿûÿ´˜¬˜Ô” ˜¨(*&
	__sig__56+¸++&Ô”¨˜ûÿÿÿ.,&The space of differentials of C1 1H1-&é3ûÿûÿûÿûÿ¼˜Ô”°˜p8.&
	__sig__57ø9:/&Ô”¸˜ûÿÿÿ€;1&CThe divisor in the divisor group D of a curve of the differential a0&4ÿûÿûÿûÿûÿÈ˜Ô”Ä˜à#	3&
	__sig__60E4&(†A vector space and the isomorphism from this space to the space of holomorphic differentials (or differentials of the first kind) of C´L5&éçÿûÿûÿûÿÜ˜Ô”Ð˜”P6&
	__sig__59¼QäQ7&Ô”Ø˜ûÿÿÿPS2&Ô”Ì˜ûÿÿÿU8& aTrue iff the divisor D is canonical, and if so also return a differential having D as its divisor@[X[9&4ûÿûÿûÿì˜Ô”à˜L`:&
	__sig__61Da\a;&Ô”è˜ûÿÿÿ|b<&0The residue of the differential a at the place Peœe´e=&þ4þÿûÿûÿûÿûÿü˜Ô”ð˜<k>&
	__sig__624lLl?&Ô”ø˜ûÿÿÿ¬m@&2The valuation of the differential a at the place PàrsA&þ4ûÿûÿûÿûÿ™Ô”™„vB&
	__sig__63ÀwØwC&Ô”™ûÿÿÿ|yU*D©P©ûÿÿÿeom/D&8/usr/local/Magma2.8/package/AlgGeom/curve/intersection.m)	+	E&8/usr/local/Magma2.8/package/AlgGeom/curve/intersection.m†p)	+	F&,True iff C and D intersect transversely at pŒG&íééûÿûÿûÿûÿ$™™™h‘I&™ ™ûÿÿÿÌ’J&
IsTransverse“ü“K&ÿÿÿÿÿÿÿÿ™þ%AreLinearlyEquivalent`ÿ%ÿÿÿÿÿÿÿÿì—&þ0ûÿûÿûÿûÿ˜˜Ô”˜<&Ô”˜ûÿÿÿ&

DivisorMap<ÛÎR;&ÿÿÿÿÿÿÿÿ˜&5The map determined by the complete linear system of Dt6&0ûÿûÿûÿûÿ,˜$˜Ô”˜d’&	__sig__48,%IsCoercibleMatrix-%ÿÿÿÿÿÿÿÿ¤”.%2/usr/local/Magma2.8/package/AlgGeom/Aux/sequence.mØ-	h7	¡0tÂ€Âûÿÿÿoc/misc.mti0)/usr/local/Magma2.8/package/RngLoc/misc.mm/`.	È.	0
lynomial1%>/usr/local/Magma2.8/package/AlgGeom/resolution/recursive_res.m	°/	0%>/usr/local/Magma2.8/package/AlgGeom/resolution/recursive_res.m	°/	¬/
__sig__0”x/	°/	4%./usr/local/Magma2.8/package/AlgGeom/curve/ff.m	h7	5%|./usr/local/Magma2.8/package/AlgGeom/curve/ff.m	h7	>%
__sig__1educible6%9True iff the curve C has exactly one place at the point p7%íéûÿûÿûÿûÿä”Ô”Ø”8%
__sig__09%Ô”à”ûÿÿÿ:%IsAnalyticallyIrreducible;%ÿÿÿÿÿÿÿÿÜ”<%:Evaluate the rational function f at the point p of a curve=%íþÿþÿûÿûÿûÿûÿ•ü”Ô”ð”?%Ô”ø”ûÿÿÿ@%?Evaluate the rational function f at degree 1 place p of a curveA%þáþÿûÿûÿûÿûÿ”Ù•Ô”•B%
__sig__2C%Ô”•ûÿÿÿE%íþÿûÿûÿûÿûÿ$••Ô”•F%
__sig__3G%Ô”•ûÿÿÿH%WThe valuation of the rational function f at the place or nonsingular point p of a curveI%þþÿûÿûÿûÿûÿ4•,•Ô” •J%
__sig__4K%Ô”(•ûÿÿÿM%íûÿûÿûÿûÿD•<•Ô”0•N%
__sig__5O%Ô”8•ûÿÿÿP%$oThe valuation of the function field of the curve on which p lies associated to the place or nonsingular point p(ÿÿÿÿÿÿÿÿ (.True iff the tangent cone to X at p is reducedÀ
	(íûÿûÿûÿûÿ, Ÿ  Ä
(
	__sig__14¬Ä(Ÿ( ûÿÿÿx(LThe singular locus of X as an scheme embedded in the same ambient space as X (ßßûÿûÿûÿûÿ< Ÿ0 ð(
	__sig__15(@(Ÿ8 ûÿÿÿ˜(SingularSubschemeØð(ÿÿÿÿÿÿÿÿ4 ("kA sequence of the singular points of X defined over K, assuming that the singular locus is zero dimensional„"œ"(ýÿßûÿûÿûÿûÿd T ŸH '(
	__sig__16(¨((ŸP ûÿÿÿ,(
SingularPoints.(ÿÿÿÿÿÿÿÿL (&|A sequence of the singular points of X defined over the base ring of X, assuming that the singular locus is zero dimensional:P:(ßûÿûÿûÿûÿl Ÿ`  =(
	__sig__17ˆ>¨>(Ÿh ûÿÿÿ@2(__sig__1Ove8=	h=	ø™	ä»ç(&|A sequence of the singular points of X defined over the base ring of X, assuming that the singular locus is zero dimensionalI¸I(ßûÿûÿûÿûÿ| Ÿp œN(Ÿx ûÿÿÿ”P ( SingularPointsOverSplittingFieldRR¨R!(ÿÿÿÿÿÿÿÿt #(:/usr/local/Magma2.8/package/AlgGeom/scheme/constructions.mu?	@@	"(:/usr/local/Magma2.8/package/AlgGeom/scheme/constructions.mp?	@@	%(>The scheme defined by all of the defining equations of X and Y	Z(
intsce&(HasSingularPointsOverExtension|c'(ÿÿÿÿÿÿÿÿ¤ ((LFalse iff all the singularities of X are defined over its current base fieldkŒk)(ßûÿûÿûÿûÿ¬ Ÿ  €n*(
	__sig__20 pèp+(Ÿ¨ ûÿÿÿ t$(ßßßûÿûÿûÿûÿ¨é´ ˆ ” ¨w,(
__sig__0xÐxøx-(ˆ ° ûÿÿÿ˜z.(
Intersection|Ä|/(ÿÿÿÿÿÿÿÿ 0(>The scheme defined by all of the defining equations of X and Yd†1(ßßßûÿûÿûÿûÿÐÇÌ ˆ À D3(ˆ È ûÿÿÿÔ4( The union of the schemes X and Y‘¸‘Ð‘5(ßßßûÿûÿûÿûÿÜ ˆ Ð ´•6(
__sig__2˜\˜,™7(ˆ Ø ûÿÿÿôš8(.The scheme X with its reduced scheme structure¬]'<pûÿÿÿ^'*The polynomial defining the hypersurface X¨_'ßKûÿûÿûÿûÿ„<x”`'
__sig__3a'<€ûÿÿÿdb'*The polynomial defining the hypersurface X c'ßKûÿûÿûÿûÿ”<ˆðd'
__sig__4(@e'<ûÿÿÿ˜f'7The hypersurface which is a common component of X and Yàøg'ßßûÿûÿûÿûÿ¤<˜´ h'
__sig__5!""i'< ûÿÿÿ$j'
CommonComponentD%k'ÿÿÿÿÿÿÿÿœl'MTrue iff the hypersurfaces X and Y do not have a common irreducible component0.m'ßßûÿûÿûÿûÿ¼<°ð5n'
__sig__67h77o'<¸ûÿÿÿà9y'ÿÿÿÿÿÿÿÿÔz'The inflection points of C= ={'éßûÿûÿûÿûÿôÌè0A|'
__sig__1BüBC}'ÌðûÿÿÿˆE'ÿÿÿÿÿÿÿÿì¸'
ponentss€'The jacobian ideal of X´LôLM'ßLûÿûÿûÿûÿžÌžäP‚'
__sig__2QüQHRƒ'ÌžûÿÿÿS„'BThe matrix of partial derivatives of the defining polynomials of XW¨W…'ßDûÿûÿûÿûÿžÌžT\†'
__sig__3]^D^‡'Ìžûÿÿÿ¼`ˆ'PThe matrix of second partial derivatives of the polynomial of the hypersurface XcDd´d‰'ßïÿûÿûÿûÿûÿ,žÌ žŒgŠ'
__sig__4jk<k‹'Ì(žûÿÿÿœlŒ'
HessianMatrix¬m'ÿÿÿÿÿÿÿÿ$žŽ'7/usr/local/Magma2.8/package/AlgGeom/scheme/firstprops.mØR	'7/usr/local/Magma2.8/package/AlgGeom/scheme/firstprops.mØR	')The number of coordinates of a point of X@zXz‘'ßûÿûÿûÿûÿLž<ž@ž8’'
__sig__0€°€È€“'<žHžûÿÿÿ†”'NumberOfCoordinatesü‡ˆ•'ÿÿÿÿÿÿÿÿDž–'7True iff the scheme X is irreducible over its base ringÈàø—'ßûÿûÿûÿûÿ¼ßdž<žXž<”˜'
__sig__1–,–D–™'<ž`žûÿÿÿšš'True iff X is reducedÔ›ô›œü&
KleinQuarticý&ÿÿÿÿÿÿÿÿä›þ&$uGiven an projective plane P2 over a finite field of q^2 elements, return the Hermitian curve x^(q+1)+y^(q+1)+z^(q+1).;;ÿ&êéûÿûÿûÿûÿœÜ›ø›d'
__sig__1á:;'Ü›œûÿÿÿ˜-;'
HermitianCurveÐ”'ÿÿÿÿÿÿÿÿü›'0/usr/local/Magma2.8/package/AlgGeom/scheme/sch.møT	PU	(p	ä»çˆ&2«Parametrization of the rational curve C given a rational point p on C. The the affine or projective line L will be set as the domain of this map if included as an argumentd‰&íéß0ûÿûÿûÿûÿLš,šô™ š‹&ô™(šûÿÿÿŒ&*True iff L is an affine or projective lined&ßûÿûÿûÿûÿ<šô™0šŽ&
__sig__3dç)I;&ô™8šûÿÿÿ‘&ÿÿÿÿÿÿÿÿ4š’&
{‹W;“&0ûÿûÿûÿûÿ\šTšô™Hšd”&
__sig__4£N;•&ô™PšûÿÿÿGÀ¦:–&*ŠParametrization of the conic curve C. The the affine or projective line L will be set as the domain of this map if included as an argumentdv'ý£:—&ß0ûÿûÿûÿûÿdšô™Xš˜&
__sig__5vªï.;™&ô™`šûÿÿÿdvš&7/usr/local/Magma2.8/package/AlgGeom/curve/singularity.m b	›&7/usr/local/Magma2.8/package/AlgGeom/curve/singularity.m b	œ&The geometric genus of C€y&éûÿûÿûÿûÿ|šlšpšž&
__sig__0{Ÿ&lšxšûÿÿÿø| &
GeometricGenusüz¡&ÿÿÿÿÿÿÿÿtš¢&#The arithmetic genus of the curve C”£&éûÿûÿûÿûÿ”šlšˆš¤&
__sig__1¥&lššûÿÿÿ)'ßûÿûÿûÿûÿ¬œœ œH+'œ¨œûÿÿÿ$,'
IsLinearScheme˜-'ÿÿÿÿÿÿÿÿ¤œ.'2True iff the ambient space of X is two dimensionalœÄ/'ßûÿûÿûÿûÿÄœœ¸œL0'
__sig__84T1'œÀœûÿÿÿœ2'
IsPlanarì3'ÿÿÿÿÿÿÿÿ¼œ4'DTrue iff Groebner basis methods are available for the equations of X€5'ßûÿûÿûÿûÿÜœœÐœ,!6'
__sig__9"€"˜"7'œØœûÿÿÿh$8'NTrue iff factorisation and irreducibility calculations can be carried out in X)9'ßûÿûÿûÿûÿìœœàœl/:'
	__sig__101D1;'œèœûÿÿÿì5<'HasFactorisation7¤7¼7='ÿÿÿÿÿÿÿÿäœ>'NTrue iff factorisation and irreducibility calculations can be carried out in X|=?'ßûÿûÿûÿûÿœøœdA@'
	__sig__110C¸CA'œûÿÿÿ´EB'HasFactorizationG<G\GC'ÿÿÿÿÿÿÿÿüœD'7True iff resultant calculations can be carried out in X˜N°NøNE'ßûÿûÿûÿûÿœ¤RF'
	__sig__12ìSTG'œûÿÿÿVH'
HasResultantW¤WI'ÿÿÿÿÿÿÿÿJ'1True iff GCD calculations can be carried out in X^@^K'ßûÿûÿûÿûÿ4œ(bL'
	__sig__13ðbxcM'œ0ûÿÿÿpeN'9/usr/local/Magma2.8/package/AlgGeom/scheme/hypersurface.m˜f	àj	O'9/usr/local/Magma2.8/package/AlgGeom/scheme/hypersurface.mðf	àj	P' _True iff X can be defined by a single equation, in which case also return the defining equation¤süs4tQ'ßKûÿûÿûÿL<@lwR'
__sig__0xŒx¤xS'<HûÿÿÿTzT'
IsHypersurface¨{U'ÿÿÿÿÿÿÿÿDV'*The polynomial defining the hypersurface X…8…W'ßKûÿûÿûÿûÿd<Xx‹X'
__sig__1$ŽLŽY'<`ûÿÿÿ\Z'*The polynomial defining the hypersurface X\’˜’['ßKûÿûÿûÿûÿDÀt<h—\'
__sig__2™è™š¦&
ArithmeticGenus\§&ÿÿÿÿÿÿÿÿŒš©&íéûÿûÿûÿ¼š¬šlš šxbª&
__sig__2c@d°d«&lš¨šûÿÿÿàe¬&IsFlex f­&ÿÿÿÿÿÿÿÿ¤š®&!True iff p is a flex on a curve Cðlm¯&íûÿûÿûÿÄšlš¸šs°&
__sig__3t´tÌt±&lšÀšûÿÿÿvÂ&
intint²&!True iff p is a flex on a curve C$yHy³&íûÿûÿûÿÔšlšÈšD~´&
__sig__4\tµ&lšÐšûÿÿÿ„‚¶&IsInflectionPointx††·&ÿÿÿÿÿÿÿÿÌš¸&
(@¹&íéûÿûÿûÿûÿüšìšlšàš”‘º&
__sig__5’°’È’»&lšèšûÿÿÿ8”¼&
IsDoublePoint@–À&
__sig__6™¨™Ð™´(;The ambient space which contains the scheme on which p lies`˜µ(íßûÿûÿûÿûÿÜ¢¢Ð¢˜·(¢Ø¢ûÿÿÿÀ¸(4/usr/local/Magma2.8/package/AlgGeom/scheme/cluster.mq	h~	¹(4/usr/local/Magma2.8/package/AlgGeom/scheme/cluster.mq	h~	º(
»(ýÿèûÿûÿûÿûÿü¢ô¢ä¢è¢ì¼(
__sig__0$<½(ä¢ð¢ûÿÿÿ”¾(=Sequence of points of Z defined over K or the base field of Z¿(èûÿûÿûÿûÿ$££ä¢ø¢° À(
__sig__1!""Á(ä¢£ûÿÿÿ$È(

rSplittingFieldÂ(6Sequence of points of Z defined over a splitting field)Ã(èõÿûÿûÿûÿD££ä¢£l/Ä(
__sig__211D1Å(ä¢£ûÿÿÿì5Æ(SupportOverSplittingField8l8Ç(ÿÿÿÿÿÿÿÿ£É(ýÿßûÿûÿûÿûÿ4£,£ä¢ £Ì=Ê(
__sig__3>,?D?Ë(ä¢(£ûÿÿÿ,AÌ(YSequence of points of the zero-dimensional scheme S defined over K or the base field of S,FDFÍ(ßûÿûÿûÿûÿ<£ä¢0£ðKÎ(
__sig__4M MÀMÏ(ä¢8£ûÿÿÿ˜OÖ(
__sig__6ingFieldÐ((ƒSequence of points of the zero-dimensional scheme S defined over a splitting field; the splitting field is returned as second valueÌVWÑ(ßõÿûÿûÿûÿL£ä¢@£Ì[Ò(
__sig__5]H]p]Ó(ä¢H£ûÿÿÿH`Ô(!Degree of the projective scheme SÐaèaÕ(ßûÿûÿûÿûÿ,«\£ä¢P£°e×(ä¢X£ûÿÿÿÐfØ(7/usr/local/Magma2.8/package/AlgGeom/scheme/basechange.mh~	Ù(7/usr/local/Magma2.8/package/AlgGeom/scheme/basechange.mh~	Ü(d£t£ûÿÿÿÞ("iThe base change of the schemes in the sequence F by coercion of the base rings or the map m of base rings|£8{	à(
__sig__7Xxá(d£€£ûÿÿÿr prç(4/usr/local/Magma2.8/package/AlgGeom/scheme/ambient.m{	°|	æ(4/usr/local/Magma2.8/package/AlgGeom/scheme/ambient.m{	°|	é(RThe product of the ambient spaces of A and B together with the two projection mapsig__5ûÿî(
__sig__0£ £ûÿÿÿê(
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unctionâ(
´‘Ì‘ã(ßßßûÿûÿûÿûÿh£”£d£ˆ£°•ä(
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__sig__3è;(ˆ è ûÿÿÿ <(
ReducedScheme`=(ÿÿÿÿÿÿÿÿä >(,The scheme X after removing any Y components of it assuming that X and Y are both hypersurfaces; the number of factors removed is also returnedœÔì?(ßßßûÿûÿûÿ¡ˆ ø @(
__sig__4A(ˆ ¡ûÿÿÿ€K(3/usr/local/Magma2.8/package/AlgGeom/scheme/points.m	HŒ	L(
D!˜!M(ííûÿûÿûÿL¡<¡,¡0¡@%N(
__sig__0'X'x'O(,¡8¡ûÿÿÿt*P(
IsRationalPoint,,Q(ÿÿÿÿÿÿÿÿ4¡R(<ÏTrue iff p lies in the base ring point set of X (or the scheme on which it lies if the second argument is omitted) or it can be coerced there. In that case, also return p coerced into the base ring point setÌ;ä;ü;S(íßíûÿûÿûÿT¡,¡H¡D?T(
__sig__1@@4@U(,¡P¡ûÿÿÿCV(=The points of the scheme X defined over its finite base field”FW(ß0ûÿûÿûÿûÿd¡,¡X¡°LX(
__sig__2MNNY(,¡`¡ûÿÿÿèOz(

cerPointon`Q[(ßûÿûÿûÿûÿ„¡t¡,¡h¡\U\(
__sig__3V|V”V](,¡p¡ûÿÿÿÌX^(HasPointsOverExtensionT[_(ÿÿÿÿÿÿÿÿl¡`(&~False iff the support of X is defined entirely over the field L, or over the base field of X if the second argument is omittedxca(ýÿßûÿûÿûÿûÿŒ¡,¡€¡Ðfb(
__sig__4gh(hc(,¡ˆ¡ûÿÿÿld(ZTrue iff X contains a nonsingular point defined over L, in which case returns such a pointœpäpe(qßíûÿûÿûÿ¬¡œ¡,¡¡vf(
__sig__5wTwlwg(,¡˜¡ûÿÿÿôxh(HasNonsingularPointlz”zi(ÿÿÿÿÿÿÿÿ”¡j($sTrue iff X contains a nonsingular point defined over the finite base field of X, in which case returns such a point¨†À†k(ßíûÿûÿûÿ´¡,¡¨¡LŽl(
__sig__6øm(,¡°¡ûÿÿÿ”‘n(4/usr/local/Magma2.8/package/AlgGeom/scheme/tangent.m‹	HŒ	o(4/usr/local/Magma2.8/package/AlgGeom/scheme/tangent.m‹	HŒ	p(
˜š°šq(íßßûÿûÿûÿûÿÜ¡Ì¡¼¡À¡ˆžÑ'ÿÿÿÿÿÿÿÿ4ŸÒ'-True iff X is nonsingular and equidimensional˜Ó'ßûÿûÿûÿûÿ¬ŸTŸŸHŸ`Ô'
__sig__2œÄÕ'ŸPŸûÿÿÿðÖ'
IsNonSingularì×'ÿÿÿÿÿÿÿÿLŸØ'-True iff X is nonsingular and equidimensional¼Ù'ßûÿûÿûÿûÿ¼ŸlŸŸ`Ÿ¼Ú'
__sig__3´ÜÛ'ŸhŸûÿÿÿŒÜ'
IsNonsingularÝ'ÿÿÿÿÿÿÿÿdŸÞ'DTrue iff X has at least one singular point or is not equidimensional%p%ß'ßûÿûÿûÿûÿŒŸ„ŸŸxŸ´+á'Ÿ€Ÿûÿÿÿü-â'
Ä/ü/ã'íûÿûÿûÿûÿœŸ”ŸŸˆŸŒ6ä'
__sig__58„8œ8å'ŸŸûÿÿÿœ:æ' _True iff p is a singularity of X, or the scheme in which it lies if no scheme argument is given>D>l>ç'íßûÿûÿûÿûÿ¤ŸŸ˜Ÿ¤Bè'
__sig__6DüDEé'Ÿ ŸûÿÿÿDF(
ty$Hê'
œI´Ië'íßûÿûÿûÿûÿÌŸ´ŸŸ¨Ÿ˜Nì'
__sig__7OPXPí'Ÿ°ŸûÿÿÿàQî'
¼RìRï'íßûÿûÿûÿûÿÜŸÄŸŸ¸ŸWð'
__sig__8X´XÌXñ'ŸÀŸûÿÿÿT[ò'"hTrue iff p is a nonsingular point of X, or of the scheme in which it lies if no scheme argument is given`a(aó'íûÿûÿûÿûÿÔŸŸÈŸàdô'
__sig__9eÈeàeõ'ŸÐŸûÿÿÿgö'"hTrue iff p is a nonsingular point of X, or of the scheme in which it lies if no scheme argument is givenmm¨m÷'íûÿûÿûÿûÿäŸŸØŸ4tø'
	__sig__10\utuù'ŸàŸûÿÿÿØvú'
¼wÔwû'íßûÿûÿûÿûÿüŸôŸŸèŸP{ü'
	__sig__11Ü}D~ý'ŸðŸûÿÿÿ¬þ'4The multiplicity of p as a point of a hypersurface X†`†ÿ'íûÿûÿûÿûÿ°¶ ŸøŸ@(
	__sig__12$X(Ÿ ûÿÿÿô(
	__sig__18ityD’(íßûÿûÿûÿûÿ$  Ÿ @–(
	__sig__13¨™Ð™(Ÿ ûÿÿÿ ›(IsOrdinarySingularity p'NoCommonComponentŒ}¤}q'ÿÿÿÿÿÿÿÿ´r'5/usr/local/Magma2.8/package/AlgGeom/scheme/jacobian.m	xœ	S)ÿÿÿÿÿÿÿÿ<¥T)5/usr/local/Magma2.8/package/AlgGeom/map/translation.m	 ©	U)
5/usr/local/Magma2.8/package/AlgGeom/map/translation.m	 ©	V)(The matrix determining f if it is linear`W)Xïÿûÿûÿûÿûÿ¼§d¥T¥X¥\Y)T¥`¥ûÿÿÿ|b)
__sig__2lex„Z)NThe translation taking the standard simplex of P to the dim(P) + 2 points of Q¼[)ÿíê0ûÿûÿûÿûÿt¥T¥h¥Œ\)
__sig__1Èè])T¥p¥ûÿÿÿØ^)TranslationOfSimplex!,!_)ÿÿÿÿÿÿÿÿl¥`)NA translation taking the points [0:..1:..0] of P to the dim(P) + 1 points of SD(a)ÿíê0ûÿûÿûÿûÿœ¥Œ¥T¥€¥ü-c)T¥ˆ¥ûÿÿÿ´0d)
Translationl2¼3e)ÿÿÿÿÿÿÿÿ„¥f)(A linear automorphism of P taking p to q:t:œ:g)ííê0ûÿûÿûÿûÿ¬¥¤¥T¥˜¥|=h)
__sig__3>¼>Ü>i)T¥ ¥ûÿÿÿ4@j)MThe automorphism which translates p to the origin (of the first affine patch)´Ek)íß0ûÿûÿûÿûÿ´¥T¥¨¥Jl)
__sig__4LðLMm)T¥°¥ûÿÿÿ˜Nn)4/usr/local/Magma2.8/package/AlgGeom/map/projection.m¡	 ©	o)4/usr/local/Magma2.8/package/AlgGeom/map/projection.m¢	 ©	p)(ƒThe affine scheme obtained by eliminating the ambient variables whose indices appear in V from the equations of the affine scheme X|[œ[q)ÿßßûÿûÿûÿûÿÌ¥¼¥À¥ˆ`r)
__sig__0apaˆas)¼¥È¥ûÿÿÿ¨bt)
EliminationÈdàdu)ÿÿÿÿÿÿÿÿÄ¥v)WThe linear projection P - -> Q which forgets the first dim(P) - dim(Q) coordinates of PHl`l˜lw)êê0ûÿûÿûÿûÿì¥ä¥¼¥Ø¥äpx)
__sig__1süs4ty)¼¥à¥ûÿÿÿ¤uz) cThe image of the projection of X away from (1:0:...:0); the ambient projection map is also returnedœyäy{)êßß0ûÿûÿûÿü¥ô¥¼¥è¥¬~|)
__sig__2ÄÜ})¼¥ð¥ûÿÿÿ8…~) cThe image of the projection of X away from (1:0:...:0); the ambient projection map is also returned(@)ßß0ûÿûÿûÿ¦¦¼¥ø¥”‘€)
__sig__3’°’È’)¼¥¦ûÿÿÿ8”‚)YThe image of the projection of X away from p; the ambient projection map is also returned›@›ƒ)íßß0ûÿûÿûÿ¦¼¥¦(Ÿ„)
__sig__4 ¸ ð r(
__sig__0ü$s(¼¡È¡ûÿÿÿHt(
TangentSpace
ì
u(ÿÿÿÿÿÿÿÿÄ¡v(JThe tangent space to S at p embedded as a scheme in the ambient space of Süw(íßûÿûÿûÿûÿä¡¼¡Ø¡¼x(
__sig__1”¬y(¼¡à¡ûÿÿÿü{(íßßûÿûÿûÿûÿü¡ô¡¼¡è¡$|(
__sig__2$D}(¼¡ð¡ûÿÿÿŒ~(GThe tangent cone at the rational point p to the scheme on which it lies@è(íßûÿûÿûÿûÿ¢¼¡ø¡$€(
__sig__3%(%@%(¼¡¢ûÿÿÿ(‚(//usr/local/Magma2.8/package/AlgGeom/scheme/pt.mX·	ƒ(//usr/local/Magma2.8/package/AlgGeom/scheme/pt.mX·	„( The origin of the affine space A3Ü3ô3…(ëíûÿûÿûÿûÿ¢¢¢Ü9†(
__sig__0:ä:ü:‡(¢¢ûÿÿÿ\<ˆ(Origin=‰(ÿÿÿÿÿÿÿÿ¢Š(EThe sequence of projective coordinate points (0,..,1,..0) and (1,..1)D‹(êûÿûÿûÿûÿ4¢¢(¢¤HŒ(
__sig__1IüIJ(¢0¢ûÿÿÿPMŽ(SimplexXN(ÿÿÿÿÿÿÿÿ,¢(:The point whose coordinates are the negative of those of pdSŒS‘(ííûÿûÿûÿûÿL¢¢@¢X’(
__sig__2YÔYZ“(¢H¢ûÿÿÿ\œ(
__sig__4]ˆ]Ð](¢p¢ûÿÿÿˆ`ž(NonZeroCoordinates a¸aŸ(ÿÿÿÿÿÿÿÿl¢ (6The indices of the zero coordinates of p as a sequencepf¡(íûÿûÿûÿûÿŒ¢¢€¢l¢(
__sig__5m mHm£(¢ˆ¢ûÿÿÿŒo¤(
ZeroCoordinatesÄr¥(ÿÿÿÿÿÿÿÿ„¢¦('True iff the points of S line on a linewTwlw§(ÿíûÿûÿûÿûÿ¤¢¢˜¢Üz¨(
__sig__6|0}d}©(¢ ¢ûÿÿÿ4ª(
AreCollinear€Ä€«(ÿÿÿÿÿÿÿÿœ¢­(ÿíßèûÿûÿûÿûÿÄ¢¼¢¢°¢ü®(
__sig__7¸Ð¯(¢¸¢ûÿÿÿd‘°(YThe cluster supported on the given point or points, inside S or the natural ambient spaceè”°•±(ÿíèûÿûÿûÿûÿÌ¢¢À¢ ›²(
__sig__8œÈœðœ³(¢È¢ûÿÿÿ¸ž¶(
__sig__9 € ˜ B(RemoveHypersurfaceœ‘C(ÿÿÿÿÿÿÿÿü D(*The scheme X with any Z components removed;E(ßßßûÿûÿûÿûÿ¡ˆ ¡¡Ú.;F(
__sig__5hšG(ˆ ¡ûÿÿÿH(

DifferenceØ›I(ÿÿÿÿÿÿÿÿ¡J(3/usr/local/Magma2.8/package/AlgGeom/scheme/points.m8¹	¹	ø™	ä»çs'
5/usr/local/Magma2.8/package/AlgGeom/scheme/jacobian.m	»	t'The inflection points of C|™u'éßûÿûÿûÿûÿÜÌÐv'
__sig__0Ø›w'ÌØûÿÿÿx'InflectionPoints¢\ 6415*x^583 - 6062*x^584 - 4696*x^585 - 2458*x^586 - 379*x^587 + 1092*x^588 + ñ\ 1231*x^589 + 371*x^590 - 1346*x^591 - 3031*x^592 - 4421*x^593 - 4837*x^594 - ° @] 4455*x^595 - 3174*x^596 - 1755*x^597 - 475*x^598 - 65*x^599 - 349*x^600 - - Ð5] 1386*x^601 - 2412*x^602 - 3237*x^603 - 3245*x^604 - 2719*x^605 - x^600 - o^ 1856*x^618 + 1803*x^619 + 731*x^620 - 1239*x^621 - 3192*x^622 - 4812*x^623 - p¾^ 5322*x^624 - 4837*x^625 - 3261*x^626 - 1425*x^627 + 308*x^628 + 1093*x^629 + "_ 993*x^630 - 172*x^631 - 1669*x^632 - 3199*x^633 - 3935*x^634 - 3959*x^635 - ´7\_ 3151*x^636 - 2251*x^637 - 1398*x^638 - 1181*x^639 - 1398*x^640 - 9ž_ 2251*x^641 - 3151*x^642 - 3959*x^643 - 3935*x^644 - 3199*x^645 - Ñ] 1680*x^606 - 917*x^607 - 568*x^608 - 1126*x^609 - 2172*x^610 - 3640*x^611 - 	hÿÿ ^ 4762*x^612 - 5376*x^613 - 4864*x^614 - 3501*x^615 - 1362*x^616 + 562*x^617 + 	è	à_ 1669*x^646 - 172*x^647 + 993*x^648 + 1093*x^649 + 308*x^650 - 1425*x^651 - ûtû/` 3261*x^652 - 4837*x^653 - 5322*x^654 - 4812*x^655 - 3192*x^656 - *q` 1239*x^657 + 731*x^658 + 1803*x^659 + 1856*x^660 + 562*x^661 - 1362*x^662 - ("À` 3501*x^663 - 4864*x^664 - 5376*x^665 - 4762*x^666 - 3640*x^667 - 2a 2172*x^668 - 1126*x^669 - 568*x^670 - 917*x^671 - 1680*x^672 - 2719*x^673 - IQa 3245*x^674 - 3237*x^675 - 2412*x^676 - 1386*x^677 - 349*x^678 - 65*x^679 - ¬I a 475*x^680 - 1755*x^681 - 3174*x^682 - 4455*x^683 - 4837*x^684 - 4421*x^685 - x^ïa 3031*x^686 - 1346*x^687 + 371*x^688 + 1231*x^689 + 1092*x^690 - 379*x^691 - œs>b 2458*x^692 - 4696*x^693 - 6062*x^694 - 6415*x^695 - 5456*x^696 - 9€b 3808*x^697 - 1725*x^698 - 111*x^699 + 818*x^700 + 548*x^701 - 478*x^702 - Ïb 2074*x^703 - 3361*x^704 - 4143*x^705 - 3931*x^706 - 3153*x^707 - x^702 - c 1937*x^708 - 1103*x^709 - 789*x^710 - 1379*x^711 - 2346*x^712 - 3541*x^713 - PR`c 4240*x^714 - 4423*x^715 - 3715*x^716 - 2558*x^717 - 1096*x^718 - 191*x^719 - ÜR¯c 52*x^720 - 1097*x^721 - 2734*x^722 - 4628*x^723 - 5844*x^724 - 6191*x^725 - ¬gþc 5260*x^726 - 3558*x^727 - 1330*x^728 + 468*x^729 + 1511*x^730 + 1227*x^731 - Ð|3,)The determinant of the Cartan matrix of G¨4,lûÿûÿûÿûÿ(÷Ø°X°Ì°Ø6,X°Ô°ûÿÿÿd7,7/usr/local/Magma2.8/package/AlgGeom/dualgraph/linking.m2	8,7/usr/local/Magma2.8/package/AlgGeom/dualgraph/linking.m2	9,/The edge determinant at the edge joining u to v°Èà:,ooûÿûÿûÿûÿð°à°ä°;,
__sig__0(<,à°ì°ûÿÿÿ€=,
EdgeDeterminant>,ÿÿÿÿÿÿÿÿè°?,The valency of u l @,oûÿûÿûÿûÿ±à°ü°<$A,
__sig__1%\%t%B,à°±ûÿÿÿH(D,ÿÿÿÿÿÿÿÿ±E,NThe root vertex of S given the directions on the edges of its underlying graphH1F,noûÿûÿûÿûÿ¨´ ±à°±p8G,
__sig__29ø9:H,à°±ûÿÿÿ€;I,

RootVertexx<<J,ÿÿÿÿÿÿÿÿ±K,;True iff S has no nodes of valency 2 and no weight 1 leavesBBL,nûÿûÿûÿûÿ8±à°,±˜FM,
__sig__3HÀHøHN,à°4±ûÿÿÿKO,@The linking number of the (virtual) link at u with the link at vO„OœOP,ooûÿûÿûÿûÿH±à°<± SQ,
__sig__4TXT€TR,à°D±ûÿÿÿ˜VS,LinkingXT,ÿÿÿÿÿÿÿÿ@±,ConnectesU,.The total linking of u with its splice diagramL`V,oûÿûÿûÿûÿ`±à°T±ÜbW,
__sig__5düdeX,à°\±ûÿÿÿDfY,
TotalLinkinggŒgZ,ÿÿÿÿÿÿÿÿX±[,&True iff S is a regular splice diagram¬m\,n
ûÿûÿûÿûÿx±à°l±8t],
__sig__6u`uxu^,à°t±ûÿÿÿÜv_,)The total linking of the root vertex in S(yLy`,nûÿûÿûÿûÿTÀˆ±à°|±H~a,
__sig__7`xb,à°„±ûÿÿÿˆ‚c,:ÊA sequence of vertices comprising the adjacent vertices on the path from u to v. The second return value is a sequence of products of those edge weights not lying on the path at the corresponding vertexàød,ooûÿûÿûÿº˜±à°Œ±<”e,
__sig__8–,–D–f,à°”±ûÿÿÿšg,

VertexPatht›¤›h,ÿÿÿÿÿÿÿÿ±i,0The euler characteristic of a generic fibre of s ¡$¡j,nûÿûÿûÿûÿ°±à°¤±L¥k,
__sig__9¦Ì¦ä¦l,à°¬±ûÿÿÿ©m,EulerCharacteristicìª«n,ÿÿÿÿÿÿÿÿ¨±„+bûÿûÿûÿûÿûÿ(®®Ì«®l…+
	__sig__25Ld†+Ì«®ûÿÿÿ”Ž+bûÿûÿûÿûÿ@®Ì«4®Ä+
	__sig__27¨À+Ì«<®ûÿÿÿà’+ÿÿÿÿÿÿÿÿ8®¥+NumberÀ“+4A sequence of numbers of possible unprojections of X”+bûÿûÿûÿûÿX®Ì«L®X•+
	__sig__28dÜ–+Ì«T®ûÿÿÿ0!—+The degree of Vœ"´"L#˜+bûÿûÿûÿûÿ€±h®Ì«\®|'™+
	__sig__29À()š+Ì«d®ûÿÿÿ0,›+%The number of X in the classification0œ+bûÿûÿûÿûÿx®Ì«l®6+
	__sig__30ˆ8 8ž+Ì«t®ûÿÿÿ :Ÿ+

Identifier¨;Ð; +ÿÿÿÿÿÿÿÿp®¡+%The number of X in the classification¨?¢+bûÿûÿûÿûÿ®Ì«„®E£+
	__sig__31FF¤+Ì«Œ®ûÿÿÿ¨H¦+ÿÿÿÿÿÿÿÿˆ®§+CThe index of X in the Altinok-Fletcher-Reid lists, 0 if not definedÔOìO¨+‹ûÿûÿûÿûÿ¸®¨®Ì«œ®PS©+
	__sig__32˜T`Uª+Ì«¤®ûÿÿÿW«+
	AFRNumber¸XÐX¬+ÿÿÿÿÿÿÿÿ ®­+CThe index of X in the Altinok-Fletcher-Reid lists, 0 if not definedt`Œ`®+bûÿûÿûÿûÿÀ®Ì«´®|c¯+
	__sig__33,eDe°+Ì«¼®ûÿÿÿtf±+-The index of X in the Reid codimension 1 list<k²+‹ûÿûÿûÿûÿ¯Ð®Ì«Ä®n³+
	__sig__34ppˆp´+Ì«Ì®ûÿÿÿ8tµ+

ReidNumber`uxu¶+ÿÿÿÿÿÿÿÿÈ®C,Valencyion·+1The index of X in the Fletcher codimension 2 listøz{¸+‹ûÿûÿûÿûÿ ¯è®Ì«Ü®à¹+
	__sig__35À‚¬„º+Ì«ä®ûÿÿÿÄ†»+
FletcherNumberÔˆ¼+ÿÿÿÿÿÿÿÿà®½+GThe index of X in the Altinok codimension 3 and 4 and exceptional listsÐ‘è‘’¾+‹ûÿûÿûÿûÿ0¯¯Ì«ô®–¿+
	__sig__36d™Œ™À+Ì«ü®ûÿÿÿD›Á+
AltinokNumber´œÂ+ÿÿÿÿÿÿÿÿø®Ã+-The index of X in the Reid codimension 1 list<¢Ä+bûÿûÿûÿûÿ¯Ì«¯t¦Å+
	__sig__37Ü§¨Æ+Ì«¯ûÿÿÿDªÇ+1The index of X in the Fletcher codimension 2 list­8­p­ˆ­`+Ì«|­ûÿÿÿa+
HilbertForm0b+ÿÿÿÿÿÿÿÿx­{+
	__sig__23neddc+NThe weights of Noether variables of X if computed, otherwise an empty sequence˜d+bûÿûÿûÿûÿ˜­Ì«Œ­Ìe+
	__sig__18àf+Ì«”­ûÿÿÿøg+NoetherFormIfAssigned¨h+ÿÿÿÿÿÿÿÿ­j+*The apparent degrees of the equations of XC	8D	i+bûÿûÿûÿûÿ¬­Ì«¨­hD	l+
	__sig__22Ð	m+
EnriquesFormä	n+ÿÿÿÿÿÿÿÿØ­o+%The weights of Noether variables of X$
p+bûÿûÿûÿûÿÈ­Ì«¼­X
q+
	__sig__20är+Ì«Ä­ûÿÿÿp!s+
NoetherFormÈ$t+ÿÿÿÿÿÿÿÿÀ­u+&The weights of Enriques variables of Xx5v+bûÿûÿûÿûÿà­Ì«Ô­Dw+
	__sig__21”Kx+Ì«Ü­ûÿÿÿ¬Kk+Ì«°­ûÿÿÿRy+)The steps in determining the weights of X|^z+bûÿûÿûÿûÿð­Ì«ä­°^|+Ì«ì­ûÿÿÿÈ^~+ÿÿÿÿÿÿÿÿè­+(A sequence of possible centres of Gorenstein projections of X together with the numbers of possible images of those projectionsXn¨v€+bûÿûÿûÿûÿ®Ì«ü­°ƒ+
	__sig__24Ü‡‚+Ì«®ûÿÿÿPŠ‘+Imagesˆƒ+ Remove the centres computed on X«è«ø*
hilbertÔfìfgû*
degrees"#$ù*
noetherìkllú*
enriquesm$mLmÿ*
unprojections$ý*
centresÈràrsþ*
deformationstÐt+

identifier#$+degree¨w+
comments#$}+Stepsoret+AFRøz{+
__sig__0#$	+

K3Database-.+
hilbert_coeffsÈ€õ* !"#$%&'()*+,-./+ _The database of numerical K3 surfaces; the string t will be the variable name of Hilbert seriesÔü+ûÿûÿûÿûÿ0¬ ¬Ì«¬,“+Ì«¬ûÿÿÿ´”
+ÿÿÿÿÿÿÿÿ¬+&|The database of numerical K3 surfaces having codimension at most n; the string t will be the variable name of Hilbert SeriesŸ\Ÿ+ûÿûÿûÿûÿ@¬8¬Ì«,¬T¤+
__sig__1¥4¥L¥+Ì«4¬ûÿÿÿD§+%The database of numerical K3 surfacesDªº*ProjectionChainsàø»*ÿÿÿÿÿÿÿÿÜª¼*ATrue iff there is a centre of Type i among the known centres of Xü—˜½*bûÿûÿûÿûÿüªlªðª´œ¾*
__sig__6ü4ž¿*lªøªûÿÿÿ\ À*
HasCentreTypeÔ¡Á*ÿÿÿÿÿÿÿÿôªÃ*bûÿûÿûÿûÿ«lª«¬¨¨,lûÿûÿûÿûÿûÿ¨²ð±œ²Hª,ð±¤²ûÿÿÿ$«, CalculateTransverseIntersections0¬,ÿÿÿÿÿÿÿÿ ²­,6/usr/local/Magma2.8/package/AlgGeom/dualgraph/rgraph.m	8b	®,D6/usr/local/Magma2.8/package/AlgGeom/dualgraph/rgraph.m	8b	-
BaseGerm-ÿÿÿÿÿÿÿÿ´-RThe space on which coordinate transformations used in the creation of G take place\-ltûÿûÿûÿûÿ ´¸²´-
	__sig__13¤-¸²´ûÿÿÿ¼	-
	BaseSpaceÐ
-ÿÿÿÿÿÿÿÿ´-
	__sig__15tionsœ-WThe number of transverse intersections of the germ used to create G with the nodes of GÔØ-lûÿûÿûÿûÿØ¶8´¸²,´¨	-
	__sig__14¼	-¸²4´ûÿÿÿÔ	-TransverseIntersectionsì	-ÿÿÿÿÿÿÿÿ0´-*The sequence of neighbouring germs along G0
-lûÿûÿûÿûÿÐ½P´¸²D´Ü-¸²L´ûÿÿÿd!-NeighbouringGerms„!-ÿÿÿÿÿÿÿÿH´-Internal functionl5-lûÿûÿûÿûÿh´¸²\´„D-
	__sig__16„K-¸²d´ûÿÿÿ K-
PrePatchMaps´K-ÿÿÿÿÿÿÿÿ`´%-
	__sig__18tion|^-JMinus the number of blowups on the x-axis in the process of constructing G°^-lûÿûÿûÿûÿÀ½€´¸²t´ä^-
	__sig__17ø^ -¸²|´ûÿÿÿ g!-BaseBlowupContribution8g"-ÿÿÿÿÿÿÿÿx´#-*The number, often referred to as delta, which is the contribution to the genus of a plane curve by a singularity having this resolution graphTŠ$-lûÿûÿûÿûÿ˜´¸²Œ´ð«&-¸²”´ûÿÿÿ¬'-GenusContributionÌ²+-
	__sig__19	ô»ç²,selfintersectionsÄÜ³,
	base_germ¼‚¨„¶,
multiplicities4´,

base_space8ˆ ˆµ,base_blowup_contribution(@¸,galois_multiplicities6·,canonical_classô‘d‘º,transverse_intersections78¹,axis_multiplicities`”€”¼,
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0¥H¥À,lûÿûÿûÿûÿûÿ³¸²ü²ªÁ,
__sig__0««X«Â,¸²³ûÿÿÿ4­Ã,HackobjPrintGrphRes¬®Ô®ü+‹ûÿûÿûÿûÿÄ¯ø¯Ì«ì¯¼
ÿ+

HasImageIn¼Ô,ÿÿÿÿÿÿÿÿÜ¯, aMake a new numerical K3 surface by adding weight w to the weights of X and recomputing other datax,bbûÿûÿûÿûÿ°Ì«°,
	__sig__53$<,Ì«°ûÿÿÿ¼,
	AddWeightÔì,ÿÿÿÿÿÿÿÿ°, eMake a new numerical K3 surface by removing weight w from the weights of X and recomputing other data,bbûÿûÿûÿûÿ(°Ì«°À#	,
	__sig__54ø$%
,Ì«$°ûÿÿÿx',
RemoveWeight(),ÿÿÿÿÿÿÿÿ °,2/usr/local/Magma2.8/package/AlgGeom/k3/make_k3db.mXf	Àq	,2/usr/local/Magma2.8/package/AlgGeom/k3/make_k3db.m¨f	Àq	,0¢A sequence of records of possible models of K3 surfaces in codimension at most maxcod with singularities from Baskets and corresponding minimum genus from MinGens¤=Ì=,ûÿûÿûÿûÿH°8°<°¼A,
__sig__0CàCD,8°D°ûÿÿÿäE,
MakeK3Database\G,ÿÿÿÿÿÿÿÿ@°¥/À”È”ûÿÿÿeom/,9/usr/local/Magma2.8/package/AlgGeom/dualgraph/numerical.m(i	Àq	,9/usr/local/Magma2.8/package/AlgGeom/dualgraph/numerical.m€i	Àq	,"iThe number of transverse intersections of the birational pullback of the base germ of G with its vertices´Z[,lûÿûÿûÿûÿh°X°\°ø_,
__sig__0`a(a,X°d°ûÿÿÿHb,NumberOfTransverseIntersectionsàd,ÿÿÿÿÿÿÿÿ`°,MThe symmetric adjacency matrix of G with selfintersections along the diagonall,lïÿûÿûÿûÿûÿ€°X°t°$p ,X°|°ûÿÿÿ¤s),
__sig__3tersection!,8The selfintersection of the total exceptional locus of Gw¼wÔw",lûÿûÿûÿûÿ°X°„°P{#,
__sig__2}Ü}D~$,X°Œ°ûÿÿÿ¬%,ExceptionalSelfIntersection…8…&,ÿÿÿÿÿÿÿÿˆ°/,__sig__4ntersectionŒdŒ', bThe intersection of the birational pullback of the base germ of G with the total exceptional locus´‘Ì‘(,lûÿûÿûÿûÿ¨°X°œ°°•*,X°¤°ûÿÿÿˆ™+,ExceptionalCurveIntersection› ›,,ÿÿÿÿÿÿÿÿ °-,"jThe degree of the exceptional part of the canonical class on the birational pullback of the base germ of Gø£ ¤.,lûÿûÿûÿûÿÀ°X°´°€§0,X°¼°ûÿÿÿª1,
CanonicalDegreeX«2,ÿÿÿÿÿÿÿÿ¸°5,
__sig__5®ì®¯È+bûÿûÿûÿûÿ(¯Ì«¯ì
É+
	__sig__38hÊ+Ì«$¯ûÿÿÿË+GThe index of X in the Altinok codimension 3 and 4 and exceptional liststœÄÌ+bûÿûÿûÿûÿ8¯Ì«,¯LÍ+
	__sig__394TÎ+Ì«4¯ûÿÿÿœÏ+The Hilbert series of X<\|Ð+bSûÿûÿûÿûÿP¯H¯Ì«<¯TÑ+
	__sig__40`€Ò+Ì«D¯ûÿÿÿÓ+TThe Hilbert series of the scheme X after cutting by hyperplane sections of degrees W##Ô+bSûÿûÿûÿûÿX¯Ì«L¯(Õ+
	__sig__41\*t*Ö+Ì«T¯ûÿÿÿ¬,×+$sThe sequence of the first n coefficients of the Hilbert series of X including the constant coefficient as the first6,6Ø+bûÿûÿûÿûÿh¯Ì«\¯œ:Ù+
	__sig__42¤;Ì;Ú+Ì«d¯ûÿÿÿì<Û+HilbertCoefficients„>¤>Ü+ÿÿÿÿÿÿÿÿ`¯Ý+ _Add at most two weights to try to make all local basket polarisations arise from global weightsDFlF”FÞ+bbûÿûÿûÿûÿ€¯Ì«t¯°Lß+
	__sig__43NNà+Ì«|¯ûÿÿÿèOá+AddLocalGeneratorsˆQ Qâ+ÿÿÿÿÿÿÿÿx¯ã+ISet the identifiers of the elements of K3 to be 1..#K3 in the given orderìWXä+ûÿûÿûÿûÿûÿ˜¯Ì«Œ¯°\å+
	__sig__44 _8_æ+Ì«”¯ûÿÿÿè`ô+
	__sig__51greesèaç+ aThe apparent degrees of equations from the initial negative coefficients of the Hilbert numerator(f@fè+‹ûÿûÿûÿûÿ¸¯¨¯Ì«œ¯èké+
	__sig__46ðlmê+Ì«¤¯ûÿÿÿ|në+ApparentEquationDegreesÄrì+ÿÿÿÿÿÿÿÿ ¯í+ aThe apparent degrees of equations from the initial negative coefficients of the Hilbert numerator$yHyî+bûÿûÿûÿûÿÀ¯Ì«´¯D~ï+
	__sig__47\tð+Ì«¼¯ûÿÿÿ„‚ò+ _The apparent codimension according to the sign changes of coefficients of the Hilbert numeratorx‹ ŒdŒñ+bûÿûÿûÿûÿÌ¯Ì«È¯ôó+Ì«Ð¯ûÿÿÿ˜’õ+ApparentCodimensionà“ø“ö+ÿÿÿÿÿÿÿÿð¯ø+^True iff there is a computed projection of Type i from X with image among the identifiers of S ÷+bûÿûÿûÿûÿä¯Ì«à¯Ì«ú+
	__sig__52sig__49ù+Ì«è¯ûÿÿÿH¥û+ _The apparent codimension according to the sign changes of coefficients of the Hilbert numerator€ª¨ªÐªý+
	__sig__50l¬„¬þ+Ì«ô¯ûÿÿÿô­®D®‡+
RemoveCentres°”ˆ+ÿÿÿÿÿÿÿÿ®‰+4Remove the centres computed on the K3 surfaces in DBœ°œŠ+ûÿûÿûÿûÿûÿ0®Ì«$®ð ‹+
	__sig__26P¢x¢Œ+Ì«,®ûÿÿÿ¸¤+*‡A sequence of possible images among DB of Gorenstein projections of X together with the numbers of possible images of those projections««X«(-ÿÿÿÿÿÿÿÿ´)-KThe root vertex of g with respect to the directions on its underlying graph¨*-lmûÿûÿûÿûÿ°´¸²¤´”,-¸²¬´ûÿÿÿH-- Nominate p as the base germ of GÜô.-ílûÿûÿûÿûÿûÿÀ´¸²´´ˆ/-
	__sig__20ˆ 0-¸²¼´ûÿÿÿ@1-
SetBaseGerm@`2-ÿÿÿÿÿÿÿÿ¸´3-@Set the multiplicities of G to be the integers of the sequence m´4-lûÿûÿûÿûÿûÿØ´¸²Ì´L#5-
	__sig__21T$l$6-¸²Ô´ûÿÿÿÔ&7-SetMultiplicities(¨(8-ÿÿÿÿÿÿÿÿÐ´9-JSet the canonical multiplicities of G to be the integers of the sequence c1¨1:-lûÿûÿûÿûÿûÿð´¸²ä´ 9;-
	__sig__22x: :<-¸²ì´ûÿÿÿ<=-SetCanonicalClass= =>-ÿÿÿÿÿÿÿÿè´Y-
	__sig__27Mapsons?-CSet the selfintersections of G to be the integers of the sequence s E¸E@-lûÿûÿûÿûÿûÿµ¸²ü´JA-
	__sig__23ôLMB-¸²µûÿÿÿœNC-SetSelfintersectionsP”PD-ÿÿÿÿÿÿÿÿµE-TSet the number of transverse intersections of G to be the integers of the sequence tVWF-lûÿûÿûÿûÿûÿ µ¸²µÐ[G-
	__sig__24L]t]H-¸²µûÿÿÿL`I-SetTransverseIntersections¤a¼aJ-ÿÿÿÿÿÿÿÿµK-6Set s to be the sequence of neighbouring germs along GtfL-lûÿûÿûÿûÿûÿ8µ¸²,µlM-
	__sig__25$mLmN-¸²4µûÿÿÿoO-SetNeighbouringGermsrsP-ÿÿÿÿÿÿÿÿ0µQ-9Set s to be the sequence of projective patch maps along Gðw xR-lûÿûÿûÿûÿûÿPµ¸²Dµ¬{S-
	__sig__26`~x~T-¸²LµûÿÿÿàU-SetProjectivePatchMaps<…V-ÿÿÿÿÿÿÿÿHµW-Internal function$ŒhŒX-lûÿûÿûÿûÿûÿhµ¸²\µøZ-¸²dµûÿÿÿœ’[-
SetPrePatchMapsÌ“\-ÿÿÿÿÿÿÿÿ`µk-
	__sig__30iessš]-9Set s to be the sequence of galois multipliticies along G$^-lûÿûÿûÿûÿûÿ€µ¸²tµ|¡_-
	__sig__28£ä£`-¸²|µûÿÿÿ¥a-SetGaloisMultiplicitiesä¦b-ÿÿÿÿÿÿÿÿxµc-7Set s to be the sequence of axis multipliticies along Gø¬­8­d-lûÿûÿûÿûÿûÿ˜µ¸²ŒµÈ°e-
	__sig__29è±²€-?@A”-ÿÿÿÿÿÿÿÿ@¶•-+The resolution graph of which v is a vertex8–-mlûÿûÿûÿûÿH¾`¶øµT¶l—-
__sig__3€˜-øµ\¶ûÿÿÿ˜™-
ResolutionGraph¬š-ÿÿÿÿÿÿÿÿX¶›-<The vertex v in the underlying graph of its resolution graphøœ-mûÿûÿûÿûÿ˜¹x¶øµl¶Äž-øµt¶ûÿÿÿÜŸ-Vertexì -ÿÿÿÿÿÿÿÿp¶¡-&The index of v in its resolution graphÈ	¢-mûÿûÿûÿûÿ¨¹¶øµ„¶ü	£-
__sig__5

¤-øµŒ¶ûÿÿÿ(
Ú-arrowsn
¥-The selfintersection of vT
¦-mûÿûÿûÿûÿ ¶øµ”¶t!§-
__sig__6$Ì$¨-øµœ¶ûÿÿÿä$©-Selfintersection5l5ª-ÿÿÿÿÿÿÿÿ˜¶«-[The multiplicity of v in the total pullback of the defining germ along its resolution graphœK¬-mûÿûÿûÿûÿÜ¿¸¶øµ¬¶R­-
__sig__7^x^®-øµ´¶ûÿÿÿ^»-
	__sig__10tions¤^¯- `The multiplicity of v in the canonical class of the germ of a surface along its resolution graph^ä^°-mûÿûÿûÿûÿÈ¶øµ¼¶(g±-
__sig__8g<g²-øµÄ¶ûÿÿÿ¸v³-CanonicalMultiplicityœƒ´-ÿÿÿÿÿÿÿÿÀ¶µ-BThe number of transverse intersections of the resolved germ with vŒ¶-mûÿûÿûÿûÿà¶øµÔ¶¬·-
__sig__9¬¼²¸-øµÜ¶ûÿÿÿØ²¼-øµì¶ûÿÿÿð²ç-
ntGrphSplÄ,ÿÿÿÿÿÿÿÿ³ß,
__sig__6phtÐtÆ,lûÿûÿûÿûÿ ³¸²³ØwÇ,
__sig__1x(yLyÈ,¸²³ûÿÿÿàz¹-BThe number of transverse intersections of the resolved germ with vðº-mûÿûÿûÿûÿð¶øµä¶Ð½-@The map from a patch on v to the base space of the defining germ`¾-m}ûÿûÿûÿûÿ·øµô¶\¿-
	__sig__114LÀ-øµü¶ûÿÿÿ„Á-ProjectivePatchMapÔìÂ-ÿÿÿÿÿÿÿÿø¶Ã-@The germs that lie on a patch of v during the resolution processtŒÄ-m}ûÿûÿûÿûÿ·øµ·èÅ-
	__sig__12D!˜!Æ-øµ·ûÿÿÿ#Ç-

PatchGerms $¸$È-ÿÿÿÿÿÿÿÿ·É-5True iff v and w are the same resolution graph vertex,,Ê-mmûÿûÿûÿûÿ¹0·øµ$·t1Ë-
	__sig__13”5ì5Ì-øµ,·ûÿÿÿ¼7Í-True iff v is a vertex of G4:L:Î-mlûÿûÿûÿûÿ@·øµ4·=Ï-
	__sig__14„>¤>Ð-øµ<·ûÿÿÿ@Ñ-<The connected component of the resolution graph containing vDEÒ-mlûÿûÿûÿûÿP·øµD·tIÓ-
	__sig__15ŒJKÔ-øµL·ûÿÿÿÀMÕ-6/usr/local/Magma2.8/package/AlgGeom/dualgraph/sgraph.m	¸®	Ö-*6/usr/local/Magma2.8/package/AlgGeom/dualgraph/sgraph.m	¸®	Ø-
edge_labels”T\UÙ-
vertex_labels”VÞ-
verticesDEÛ-corresponding_graph´Z[Ü-corresponding_vertices°\ß-
is_regulariplicitiesGÝ-linking_numbersè`a(aá-
BC/.Arrowsmà-
arrow_weightsàd×-0BCDEFG78HIJâ-nûÿûÿûÿûÿûÿ·X·„·Hlã-
__sig__0m`mxmä-X·Œ·ûÿÿÿ¼oå-HackobjPrintGrphSpl,sdsæ-ÿÿÿÿÿÿÿÿˆ·ï-
__sig__2SplvØvè-nûÿûÿûÿûÿ¨·X·œ·Tzé-
__sig__1{x{¨{ê-X·¤·ûÿÿÿ¬~ë-HackobjCoerceGrphSpl€ì-ÿÿÿÿÿÿÿÿ ·í-=The splice diagram on graph g with edge labels e and arrows adŒî-bnûÿûÿûÿûÿÐ·À·X·´·ôð-X·¼·ûÿÿÿ˜’ñ-MakeSpliceDiagramà“ø“ò-ÿÿÿÿÿÿÿÿ¸·ó-JThe splice diagram with edges corresponding to the elements of e (sequences of vertex numbers), which have edge labels given by the elements of l; and with arrows (all of weight 1) corresponding to a (the ith element of a becomes the number of arrows of the ith vertex)¸¤ô-ÿÿÿnûÿûÿûÿûÿØ·X·Ì·¨¨õ-
__sig__3ª`ª€ªö-X·Ô·ûÿÿÿ„¬÷-BThe reduced splice diagram corresponding to the resolution graph G¬¯Ô¯ø-lnûÿûÿûÿûÿø·è·X·Ü·<³Ï,MakeResolutionGraphÌ~Ð,ÿÿÿÿÿÿÿÿ0³Ò,blûÿûÿûÿûÿX³P³¸²D³ ‰Ó,
__sig__3(@Ô,¸²L³ûÿÿÿÐÕ,j‹The resolution graph supported on g. The list 'base' can contain a base germ and a base selfintersection contribution; the list L contains the following sequences of data associated to the vertices of g as follows: s = selfintersections, germ pullback multiplicities, canonical multiplicities, galois multiplicities, t = transverse intersections, projective patch maps, birational pullback germs@¡x¡Ö,ÅÅblûÿûÿûÿûÿØµ`³¸²T³¥×,
__sig__4¦§@§Ø,¸²\³ûÿÿÿ€©Ù,The underlying graph of Gˆ« «Ú,lbûÿûÿûÿûÿ¸p³¸²d³„¯Û,
__sig__5°¬°Ä°Ü,¸²l³ûÿÿÿT²ù-
__sig__4`ˆú-X·ä·ûÿÿÿœû-
SpliceDiagram
ü-ÿÿÿÿÿÿÿÿà·ý-RThe reduced splice diagram corresponding to the graph G but without reduction at vŒ´þ-mlnûÿûÿûÿûÿ¹¸X·ô·ÿ-
__sig__5h€.X·ü·ûÿÿÿÀ.The underlying graph of Sh€.nbûÿûÿûÿûÿ¸X·¸€.
__sig__6x.X·¸ûÿÿÿX.The number of vertices of SDì.nûÿûÿûÿûÿ ¸X·¸$.
__sig__7%,%D%.X·¸ûÿÿÿ(	.The labels on the edges of S*ð*
.nûÿûÿûÿûÿ0¸X·$¸¸0.
__sig__81p2À3.X·,¸ûÿÿÿ6.&yThe labels on the vertices of S, by default the arrows of S but also linking or canonical class information if calculatedH<`<.nûÿûÿûÿûÿ@¸X·4¸¨?.
__sig__9AHAhA.X·<¸ûÿÿÿ D.	__sig__12cesnGraphEèE.*The resolution graph that corresponds to S0IxI.nûÿûÿûÿûÿP¸X·D¸\N.
	__sig__10ÔOìO.X·L¸ûÿÿÿ¤Q.CorrespondingResolutionGraphSPS.ÿÿÿÿÿÿÿÿH¸.LThe vertices of the resolution graph of S corresponding to the vertices of SZ[.nûÿûÿûÿûÿh¸X·\¸ü_.
	__sig__11a,a.X·d¸ûÿÿÿLb.CorrespondingVertices´d.ÿÿÿÿÿÿÿÿ`¸.$The linking numbers at vertices of Sh,h.nûÿûÿûÿûÿ€¸X·t¸¬m .X·|¸ûÿÿÿ(p!.
LinkingNumbershs".ÿÿÿÿÿÿÿÿx¸#. _The canonical multiplicities at vertices of S of corresponding vertices in the resolution graph|y yèy$.nûÿûÿûÿûÿ˜¸X·Œ¸°~%.
	__sig__13Èà&.X·”¸ûÿÿÿ<…-.
	__sig__15es†Ä†'.ZThe galois multiplities at vertices of S of corresponding vertices in the resolution graphøŽ(.nûÿûÿûÿûÿ¨¸X·œ¸œ’).
	__sig__14”“Ì“*.X·¤¸ûÿÿÿ–+.,The arrows at vertices of the splice diagramš´š,.nûÿûÿûÿûÿÐ¹¸¸X·¬¸Œž..X·´¸ûÿÿÿœ 0.ÿÿÿÿÿÿÿÿ°¸O.
rowsA1.;The weights of the arrows at vertices of the splice diagramÜ§¨2.nûÿûÿûÿûÿÐ¸X·Ä¸ˆ¬3.
	__sig__16 ­¸­4.X·Ì¸ûÿÿÿˆ¯5.
ArrowWeights°È°6.ÿÿÿÿÿÿÿÿÈ¸É,HackobjCoerceGrphRes¥L¥Ê,ÿÿÿÿÿÿÿÿ³Ë,
,©„©Ì,blûÿûÿûÿûÿH³8³¸²,³ˆ­Í,
__sig__2®°®Ø®Î,¸²4³ûÿÿÿ˜°Ñ,

h± ±¸±Ð±w,
	__sig__11nityLo,5The milnor number at infinity of a generic fibre of Plp,rûÿûÿûÿûÿÈ±à°¼±ôq,
	__sig__10`xr,à°Ä±ûÿÿÿÄs,MilnorNumberAtInfinityðt,ÿÿÿÿÿÿÿÿÀ±u,6True iff s has a vertex with zero total linking numberv,nûÿûÿûÿûÿà±à°Ô±ðx,à°Ü±ûÿÿÿLy,HasIrregularFibres@z,ÿÿÿÿÿÿÿÿØ±{,6/usr/local/Magma2.8/package/AlgGeom/dualgraph/modify.m	xÔ	|,6/usr/local/Magma2.8/package/AlgGeom/dualgraph/modify.m	xÔ	},PConnect the resolution graphs containing v and w together by an edge from v to w-è-.~,mmlûÿûÿûÿûÿ²ð±ô±Ð4,
__sig__06è607€,ð±ü±ûÿÿÿ 9‚,ÿÿÿÿÿÿÿÿø±ƒ,.›Remove the edge of the resolution graph containing both u and w by cutting the edge between them. The resulting graph may well have more than one componentˆAÀA„,mmlûÿûÿûÿûÿ²ð±²HF…,
__sig__1HhH¨H†,ð±²ûÿÿÿxJ‡,

Disconnect4MTMˆ,ÿÿÿÿÿÿÿÿ²‰,QThe resolution graph G with an edge between the i-th and j-th vertices of G added8SPSŠ,llûÿûÿûÿûÿ0²ð±$²¨W‹,
__sig__2YHYYŒ,ð±,²ûÿÿÿÐ[,:The resolution graph G with the edges specified in E added4`L`Ž,ÿllûÿûÿûÿûÿ@²ð±4²Üb,
__sig__3düde,ð±<²ûÿÿÿDf‘,&The subgraph of G on the vertices of V,h’,ÿmllûÿûÿûÿûÿP²ð±D²¬m“,
__sig__4o¨oÀo”,ð±L²ûÿÿÿhs•,
Subgraphtètu–,ÿÿÿÿÿÿÿÿH²¯,graphØw—,Remove the i-th vertex from Gèy˜,lûÿûÿûÿûÿûÿh²ð±\²°~™,
__sig__5Èàš,ð±d²ûÿÿÿ<…©,
__sig__8citiesÄ†›,JCalculate the canonical class on the exceptional curves corresponding to G(ŽPŽœ,lûÿûÿûÿûÿûÿx²ð±l²’,
__sig__6’“,“ž,ð±t²ûÿÿÿ´”Ÿ,CalculateCanonicalClass,™ ,ÿÿÿÿÿÿÿÿp²¡,ICalculate the multiplicities of the pullback of the germ used to create GDŸ\Ÿ¢,lûÿûÿûÿûÿûÿ²ð±„²T¤£,
__sig__7¥4¥L¥¤,ð±Œ²ûÿÿÿD§¥,CalculateMultiplicities„©¦,ÿÿÿÿÿÿÿÿˆ²±,labelsrseI§,+Calculate the transverse intersections of G0°P°k0‘‘ûÿûÿûÿûÿÄÁ´ÁÀ¨ÁHm0À°Áûÿÿÿ$n0

LocalField€˜o0ÿÿÿÿÿÿÿÿ¬Áp0>The degree f unramified extension of L, reduced to precision nq0‘‘ûÿûÿûÿûÿÔÁÌÁÀÀÁ|r0
	__sig__24l„s0ÀÈÁûÿÿÿìt0,The totally ramified extension of L defined by the Eisenstein polynomial g or the unramified extension of L defined by the inertia polynomial gÜôu0‘‘ûÿûÿûÿûÿäÁÜÁÀÐÁ° v0
	__sig__25""w0ÀØÁûÿÿÿ$x04°The totally ramified extension of L defined by the Eisenstein polynomial g, reduced to precision n or the unramified extension of L by the ienrtia polynomial g with precision n-.,.y0‘‘ûÿûÿûÿûÿôÁìÁÀàÁì5z0
	__sig__26d7Œ7{0ÀèÁûÿÿÿÜ9|04The unramified local field with prime p and degree f<,<}0‘ûÿûÿûÿûÿÂüÁÀðÁt?~0
	__sig__27A,A0ÀøÁûÿÿÿ¸C€0AThe unramified local field with prime p, degree f and precision n<G\G0‘ûÿûÿûÿûÿÂÂÀÂPM‚0
	__sig__28pN˜Nƒ0ÀÂûÿÿÿP„0.˜The totally ramified local field with prime p defined by the Eisenstein polynomial g or the unramified local field with prime p and inertia polynomial gWTWtW…0‘ûÿûÿûÿûÿ$ÂÂÀÂ\†0
	__sig__29ˆ]Ð]‡0ÀÂûÿÿÿˆ`ˆ04µThe totally ramified local field with prime p and precision n defined by the Eisenstein polynomial g or the unramified local field with prime p, inertia polynomial g and precision nÐf‰0‘ûÿûÿûÿûÿ4Â,ÂÀ Â˜lŠ0
	__sig__30m¨m‹0À(Âûÿÿÿ$pŒ0XThe local field with prime p and inertia degree f defined by the Eisenstein polynomial guìuv0‘ûÿûÿûÿûÿDÂ<ÂÀ0ÂxyŽ0
	__sig__31ÄzÜz0À8Âûÿÿÿ¤}0 eThe local field with prime p, inertia degree f and precision n defined by the Eisenstein polynomial g†‘0‘ûÿûÿûÿûÿTÂLÂÀ@Â’0
	__sig__32ôŽ“0ÀHÂûÿÿÿÄ”0$rThe local field with prime p defined by the polynomials f and g, one of which is inertial and the other Eisensteinè•–•0ûÿûÿûÿûÿdÂ\ÂÀPÂÐ›–0
	__sig__33 —0ÀXÂûÿÿÿ(Ÿ˜0(ƒThe local field with prime p defined by the polynomials f and g, one of which is inertial and the other Eisenstein with precision nx¥¥™0ûÿûÿûÿûÿlÂÀ`Â@ªš0
	__sig__34ˆ« «›0ÀhÂûÿÿÿ„­+4-/usr/local/Magma2.8/package/ModSym/analytic.m	€ä	b4,/home/was/magma/packages/modsym/code/arith.m+œ07/usr/local/Magma2.8/package/RngLoc/LocalAutomorphisms.mØå	07/usr/local/Magma2.8/package/RngLoc/LocalAutomorphisms.mØå	ž0^Return the automorphisms of the local ring L. Automorphisms are returned as a sequence of maps¼¤0
__sig__1¼ô¼½z/H½ä½ûÿÿÿŒ
{/DualGraphMultiplicitiesÔ|/ÿÿÿÿÿÿÿÿà½}/$tThe sequence of multiplicities of exceptional curves in the canonical class along the first toric neighbourhood of pð~/íûÿûÿûÿûÿ¾H½ô½„€/H½ü½ûÿÿÿì/DualGraphCanonical\|‚/ÿÿÿÿÿÿÿÿø½ƒ/TThe sequence of selfintersections of exceptional curves in the resolution spine of p¼„/íûÿûÿûÿûÿ¾H½¾h"…/
__sig__9#Ø#$†/H½¾ûÿÿÿp%‡/7/usr/local/Magma2.8/package/AlgGeom/resolution/blowup.m˜ö	ˆ/7/usr/local/Magma2.8/package/AlgGeom/resolution/blowup.m˜ö	‰/,’The birational automorphism of P determined by the matrix of natural numbers M: the rows are the powers appearing in the images of the coordinatesä89Š/ïÿêûÿûÿûÿûÿ0¾ ¾$¾\<‹/
__sig__0=T=|=Œ/ ¾,¾ûÿÿÿD?/ToroidalAutomorphismA,AŽ/ÿÿÿÿÿÿÿÿ(¾/;/usr/local/Magma2.8/package/AlgGeom/resolution/resolution.mÀë	˜ö	/;/usr/local/Magma2.8/package/AlgGeom/resolution/resolution.mì	˜ö	‘/
NN’/íélûÿûÿûÿûÿX¾P¾@¾D¾àQ“/
__sig__0RSS”/@¾L¾ûÿÿÿ\U•/MThe dual graph of the minimal transverse resolution of the singularity p of CŒY–/élûÿûÿûÿûÿ¬¾`¾@¾T¾@^—/
__sig__1`p`ˆ`˜/@¾\¾ûÿÿÿ¸a™/:/usr/local/Magma2.8/package/AlgGeom/resolution/spine_res.mxî	˜ö	š/:/usr/local/Magma2.8/package/AlgGeom/resolution/spine_res.mÐî	˜ö	›/$tThe initial linear graph, the spine, and the associated data of the resolution of the germ p and its associated datannœ/ílûÿûÿûÿûÿx¾h¾l¾Ìt/
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ResolutionSpine¤x /ÿÿÿÿÿÿÿÿp¾¡/ cThe polynomial f with all i-th variable factors removed; also returns the number of factors removed¬€Ä€¢/KKûÿûÿûÿ¾h¾„¾ ˆ£/
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__sig__8à	z.(¹ä¹ûÿÿÿ¨{.?The sequence of near and far labels on the edge joining u and v$<T|.ooûÿûÿûÿûÿø¹(¹ì¹”}.
__sig__9x~.(¹ô¹ûÿÿÿ°.?True iff u and v have the same index in a common splice diagramPh€€.ooûÿûÿûÿûÿäÄº(¹ü¹€.
	__sig__10x‚.(¹ºûÿÿÿXƒ.:/usr/local/Magma2.8/package/AlgGeom/dualgraph/misc_graph.mÀù	P	„.:/usr/local/Magma2.8/package/AlgGeom/dualgraph/misc_graph.mú	P	….$tA sequence of vertices comprising the adjacent vertices on the path from u to v. G is assumed to be a directed graph+,†.ûÿûÿûÿûÿ ºººH1‡.
__sig__034Ð4ˆ.ººûÿÿÿ7‰.0ŸThe sequence of branching vertices --- those of valency at least 3 --- on the path from u to v; u and v are included as first and last elements of the sequenceÐ=ø= >Š.ûÿûÿûÿûÿ0ºº$ºB‹.
__sig__1DPDÈDŒ.º,ºûÿÿÿF.BranchVertexPathHhH¨HŽ.ÿÿÿÿÿÿÿÿ(º.0¡True if and only if G is a tree and for some vertex v, given any vertex w there exists a directed path from v to w. In this case, also return v as a second valueS S.bûÿûÿûÿHºº<ºxW‘.
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__sig__4x(yLyž.ºtºûÿÿÿàzŸ.The line graph on n vertices}H~ .bûÿûÿûÿûÿˆºº|º<…¡.
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__sig__7¨ü¨©®.º´ºûÿÿÿ«°.ÿÿÿÿÿÿÿÿ°º±.-True iff v is the root of a known rooted tree°².ûÿûÿûÿûÿÐººÄº°³³.
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GenericGenus,/ÿÿÿÿÿÿÿÿ ¼-/The set of singular fibres of LÌÐ./ûÿûÿûÿûÿÀ¼H»´¼œ//
	__sig__13°0/H»¼¼ûÿÿÿÈ1/
SingularFibresÜ2/ÿÿÿÿÿÿÿÿ¸¼3/&The sequence of nonreduced fibres of L¸	4/ûÿûÿûÿûÿØ¼H»Ì¼ì	5/
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	__sig__16esÜ9/1True iff all (finite) fibres of L are nonsingularx!:/ûÿûÿûÿûÿð¼H»ä¼`5;/
	__sig__15t5</H»ì¼ûÿÿÿdD=/HasNonSingularFibresˆD>/ÿÿÿÿÿÿÿÿè¼?/-True iff all (finite) fibres of L are reduced¼K@/ûÿûÿûÿûÿ½H»ü¼„^B/H»½ûÿÿÿœ^C/HasReducedFibres^´^D/ÿÿÿÿÿÿÿÿ½E/*True iff the fibre f = a of L is connectedø^F/þÿûÿûÿûÿûÿ ½H»½<gG/
	__sig__17´vH/H»½ûÿÿÿ¸I/IsConnectedFibreƒ´ƒJ/ÿÿÿÿÿÿÿÿ½U/
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	__sig__18ä²N/H»4½ûÿÿÿüµO/IrreducibleComponentsl·P/ÿÿÿÿÿÿÿÿ0½7.The vertices of SŒu¤u8.nûÿûÿûÿûÿè¸X·Ü¸ôx9.
	__sig__17<zTz:.X·ä¸ûÿÿÿÀ|;.>Set the numbers arrows on vertices of S to be the entries of a€<.nûÿûÿûÿûÿûÿø¸X·ì¸ˆ=.
	__sig__18@‹x‹>.X·ô¸ûÿÿÿ¼Ž?.
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	__sig__20È¦à¦H.X·¹ûÿÿÿ©I.7/usr/local/Magma2.8/package/AlgGeom/dualgraph/svertex.mÀ#	J.7/usr/local/Magma2.8/package/AlgGeom/dualgraph/svertex.mÀ#	K.
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__sig__35´Bh/H½œ½ûÿÿÿ€Di/ProjectiveRepresentativeKŒKj/ÿÿÿÿÿÿÿÿ˜½k/The base ring of pRl/íýÿûÿûÿûÿûÿLÆ¸½H½¬½Œ^n/H½´½ûÿÿÿ¤^×$
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	__sig__31r-¸²Äµûÿÿÿs-ModifySelfintersectiont-ÿÿÿÿÿÿÿÿÀµu-'The resolution graph corresponding to Nv-=lûÿûÿûÿûÿàµ¸²Ôµx-¸²Üµûÿÿÿy-#True iff G and H are the same graphz-llûÿûÿûÿûÿ(·ðµ¸²äµ{-
	__sig__33|-¸²ìµûÿÿÿ}-7/usr/local/Magma2.8/package/AlgGeom/dualgraph/rvertex.mhE	~- 7/usr/local/Magma2.8/package/AlgGeom/dualgraph/rvertex.mhE	-resolution_graph <	hE	´/?The resolution graph at infinity of the plane jacobian pencil P˜ðµ/þÿþÿlûÿûÿûÿûÿÜ¾¤¾Ð¾Ð¸//The set of vertices of g having arrows attached¨À¹/lûÿûÿûÿûÿŒ¿ì¾¤¾à¾ðº/
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__sig__6lVerticesÄ/JThe set of horizontal vertices of g which meet strongly irregular vertices/l/Å/lûÿûÿûÿûÿ¬¿¿¤¾¿\6Æ/
__sig__578l8Ç/¤¾¿ûÿÿÿL:È/StronglyHorizontalVerticesä;ü;É/ÿÿÿÿÿÿÿÿ¿Ê/MThe univariate polynomials in S of P at horizontal vertex i in the resolution¼AË/ûÿûÿûÿûÿ4¿¤¾(¿DFÍ/¤¾0¿ûÿÿÿôHÎ/HorizontalFunctionŒJKÏ/ÿÿÿÿÿÿÿÿ,¿Ð/WIrregular values of P appearing at horizontal vertex of index i in the resolution graphtRŒR¤RÑ/ûÿûÿûÿûÿL¿¤¾@¿”VÒ/
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IrregularValuesP\Õ/ÿÿÿÿÿÿÿÿD¿Þ/
__sig__9Values(aÖ/KThe finite values of the generating polynomial of P on its resolution graphÈdàd×/ûÿûÿûÿûÿd¿¤¾X¿ÈgØ/
__sig__8k`kˆkÙ/¤¾`¿ûÿÿÿÈlÚ/StronglyIrregularValuesnÛ/ÿÿÿÿÿÿÿÿ\¿Ü/BThe pullback of f:z^d to the vertex i of the resolution graph of P,vDvÝ/KKûÿûÿûÿ|¿¤¾p¿äyß/¤¾x¿ûÿÿÿP{à/
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	__sig__104\å/¤¾¿ûÿÿÿÌ‘æ/BThe set of initial irregular vertices of the resolution graph of P˜”°”ç/ûÿûÿûÿûÿ¤¿¤¾˜¿@›è/
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	__sig__12ˆ« «í/¤¾°¿ûÿÿÿ„­î/6/usr/local/Magma2.8/package/AlgGeom/resolution/ndual.m	ØM	ï/6/usr/local/Magma2.8/package/AlgGeom/resolution/ndual.m	ØM	ð/?A sequence of primitive points on the rays of the dual fan to Nlµ„µœµñ/=ûÿûÿûÿûÿÌ¿¼¿À¿4ºò/
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	__sig__15 M0À0ÁûÿÿÿN0$qThe local ring with prime p defined by the polynomials f and g, one of which is inertial and the other EisensteinO0ûÿûÿûÿûÿLÁDÁÀ8ÁäP0
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__sig__7žPžˆž-0À°Àûÿÿÿ˜ .0,The totally ramified extension of L defined by the Eisenstein polynomial g or the unramified extension of L defined by the inertia polynomial g@§h§€§/0ûÿûÿûÿûÿÌÀÄÀÀ¸À$¬00
__sig__8­l­„­10ÀÀÀûÿÿÿ¯204°The totally ramified extension of L defined by the Eisenstein polynomial g, reduced to precision n or the unramified extension of L by the inertia polynomial g with precision nµ„µœµ30ûÿûÿûÿûÿÜÀÔÀÀÈÀ4º40
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BrandtModule“,“1ÿÿÿÿÿÿÿÿôÃ12The Eichler-Brandt quaternion ideal divisor group.Üšôš1²ûÿûÿûÿûÿÄÄÃÄ¼ž1
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__sig__3´È´à´1Ã0Äûÿÿÿ8¶17The Eichler-Brandt quaternion ideal divisor group of G.€¹È¹8º1‰²ûÿûÿûÿûÿLÄDÄÃ8Ä½1
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	__sig__11ûÿûÿûÿûÿý5äÖø×ûÿÿÿ__si6Returns true if x is in X.	¼ç6
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ModularDegreeœ•4ÿÿÿÿÿÿÿÿ(Ò—4<ÏThe congruence number r of M. This is the index in S=S_k(Gamma_0(N),Z) of the sum L+W of the lattice of cusp form L corresponding to M and the lattice of cusp forms corresponding to the complement of L in S.rder of the ¥4
	__sig__10 r(M), whe¤4¨Ñ”Òûÿÿÿ1(X_§4>ÚAn abelian group G that encodes information about the intersection of the collection of complex tori corresponding to the sequence S of spaces of modular symbols. When G is finite, G is isomorphic to this intersection.¦4“ûÿûÿûÿûÿ Ò¨ÑœÒ%š4pNewModularDegree\'|'›4ÿÿÿÿÿÿÿÿ@Òœ4¶“ûÿûÿûÿûÿlÉxÒ¨ÑtÒm L+£48ÀAn abelian group G that encodes information about the intersection of the complex tori corresponding to M1 and M2. When the intersection group is finite, it is isomorphic to this intersection.ÿ¢4¶¶“ûÿûÿûÿûÿ˜ÒÒ¨ÑŒÒH? 4CongruenceModulusA0A¡4ÿÿÿÿÿÿÿÿXÒ©4
	__sig__11ence group¨4¨Ñ¤Òûÿÿÿ spa¬4¶ÿÿûÿûÿûÿûÿ¸Ò¨Ñ´Òo M,Ý4zËThe Hecke submodule of the vector space Phi(M) generated by the image of the jth eps-twisted modular winding element, where Phi is RationalMapping(M). This module is useful, for example, because in characteristic 0, if M is new of weight 2, has sign +1 or -1, and corresponds to a collection {f_i} of Galois-conjugate newforms, then the dimension of the twisted winding submodule equals the cardinality of the subset of those f_i such that L(f_i,eps,1) =/= 0.¶ã4zÊThe rational number L(A,j)*(j-1)! / ((2pi)^(j-1)*Omega), where j is a "critical integer", so 1<=j<=k-1, and Omega is the volume of A(R) when j is odd, and the volume of the -1 eigenspace for conjugation when j is even. The volume is computed with respect to a saturated basis of cusp forms that have integral Fourier coefficients at infinity. If the optional parameter Bound is set, then the result is a divisibility upper bound on the above rational number.h	(	ä4¨Ñ”Óûÿÿÿ5
rent¨xª4IntersectionGroup@zXz«4ÿÿÿÿÿÿÿÿˆÒ­4X@The order generated by the Fourier coefficients of one of the q-expansions of a newform corresponding to M, along with a map from the ring containing the coefficients of qExpansion(M) to the ring. If the optional integer parameter Bound is set, then the algebra generated only by those T_n, with n <= Bound, is computed.Ñ°ÒÐ‘¯4
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	__sig__15entrated ±4:ÌThe number field generated by the Fourier coefficients of one of the q-expansions of a newform corresponding to M, along with a map from the ring containing the coefficients of qExpansion(M) to the field.then th°4¶õÿûÿûÿûÿÈÒ¨ÑÄÒ³4
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	__sig__16entÐ±·4EThe element sum_{a in (Z/mZ)^*} <eps(a)*X^(i-1)*Y^(k-2-i-1)*{0,a/m}>.-expan¶4¯¶·ûÿûÿûÿûÿàÒ¨ÑÜÒ qEx¸4¨ÑäÒûÿÿÿ½4The winding element {0,oo}.õÿ¼4¶·ûÿûÿûÿûÿÓøÒ¨ÑôÒûÿÿÿº4HeckeEigenvalueFieldÄÄÄ»4ÿÿÿÿÿÿÿÿÀÒ¥2¬¸ûÿûÿûÿûÿ¤ÊœÊÐò˜Ê¨Ê§2Ðò Êûÿÿÿª2;The space of weight k modular forms on G over the integers.space of©2¬¸ûÿûÿûÿûÿ´Ê¬ÊÐò¨Ê¬2
__sig__4ÿ¸Ê«2Ðò°Êûÿÿÿ®2,–The direct sum of the spaces ModularForms(eps,k), where eps runs through representatives of the Galois orbits of the characters in the sequence chars.­2ÿ¯¸ûÿûÿûÿûÿ¼ÊÐò¸Ê°2
__sig__5Ê¼Êûÿÿÿ²2Coerce x into M.he direct s±2¸¹ûÿûÿûÿÌÊÐòÈÊ the´2
__sig__6hars.œp;³2ÐòÐÊûÿÿÿ¶2LThe base extension of M to the commutative ring R and the induced map from M to BaseExtend(M,R). The only requirement on R is that there is a natural coercion map from BaseRing(M) to R. For example, when BaseRing(M) is the integers then any commutative ring R is allowed.À	˜	µ2ýÿ¸¸ûÿûÿûÿìÊÜÊÐòØÊmÂ1ÐÆÇûÿÿÿdÃ1CuspidalSubspaceˆdÄ1ÿÿÿÿÿÿÿÿÇÅ1ZReturns true if and only if M is contained in the cuspidal subspace of its abmient module.—¸Ã:Æ1²²ûÿûÿûÿûÿÔÍ ÇÐÆÇdvÇ1
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__sig__4ÔxÎ1ÐÆ4ÇûÿÿÿÏ19Decomposition of M with respect to the matrix operator T.Ð1²ûÿûÿûÿûÿhÕHÇÐÆ<Çø|Ñ1
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__sig__6í];Ö1ÐÆTÇûÿÿÿ[;×1IsIndecomposableØ1ÿÿÿÿÿÿÿÿPÇÙ1CSort the sequence of Brandt modules according to the 'lt' operator.”ƒÚ1ÿ²ûÿûÿûÿûÿ¼ÕpÇÐÆdÇÛ1
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MinusVolume$XF4ÿÿÿÿÿÿÿÿôÐH4.˜The value L(M,j), where j is a critical integer, i.e., one of the integers 1,2,...,k-1. Series are are computed to precision n. The sign of M must be 0.¶G4¶VûÿûÿûÿûÿXò$ÑDó Ñ8.	J4
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__sig__7t be 0.¦M4Dó8ÑûÿÿÿR4.The number r_j(f) = int_{0}^{ioo} f(z) z^j dz.	°/	T4
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__sig__0 is a newfoÃ<”ÌœÌûÿÿÿeben03
__sig__1 that is, u/3”ÌÀÌûÿÿÿter 23>The characteristic polynomial of the n-th Hecke operator on M.¯ûÿ13¸ûÿûÿûÿûÿÄÓÌÌ”ÌÈÌûÿÿÿ)3DirichletCharacter`”€”*3ÿÿÿÿÿÿÿÿ„Ì+3./usr/local/Magma2.8/package/ModFrm/operators.m	@Å	63./home/was/magma/packages/modform/code/p-adic.m	@Å	43
__sig__2aluenting .3LThe eigenvalue of the Atkin-Lehner involution W_q on the cuspidal newform f.-3¹þÿûÿûÿûÿûÿ¼Ì”Ì¸Ì°ÌÆ<dÏlÏûÿÿÿ/cod33”ÌÐÌûÿÿÿkin-,3:ÌThe slopes of the Newton polygon of the characteristic polynomial f of the Hecke operator T_p acting on M, where M is defined over pAdicField(p). This is a list of the valuations of the p-adic roots of f.Ìûÿÿÿ83¸ÅûÿûÿûÿûÿóÔÌ°ÌÔ¸73
__sig__0Ä<ÔÌÜÌûÿÿÿÜÓ?3ÔÌÍûÿÿÿûÿÿÿ93
HeckePolynomial8¿:3ÿÿÿÿÿÿÿÿÄÌ;3+/usr/local/Magma2.8/package/ModFrm/p-adic.mÈÄ	@Å	@3__sig__1ËÜËüË,ÌDÌ3NumberOfNewformClasses3ÿÿÿÿÿÿÿÿÌ34The first Galois-conjugate newform in the ith orbit.3Ì@Ìûÿÿÿã24ËËûÿÿÿˆÆ	¿<
__sig__0å2HackobjPrintModFrmEltæ2ÿÿÿÿÿÿÿÿlËè2Name the ith generator.ç2¸#ûÿûÿûÿûÿÌ×¤Ë4Ë Ëë20/home/was/magma/packages/modform/code/l_series.mÈ	˜Ç	°È	Ú2,/home/was/magma/packages/modform/code/misc.mÇ	°È	î2,/home/was/magma/packages/modform/code/misc.mÈ	°È	À<¸Ëüòûÿÿÿï2+/usr/local/Magma2.8/package/ModFrm/level1.m°È	hÌ	ð2-/usr/local/Magma2.8/package/ModFrm/l_series.m	hÌ	ñ2)/usr/local/Magma2.8/package/ModFrm/misc.ms.@É	hÌ	ô27/home/was/magma/packages/modform/code/modular_symbols.m°Ê	ó27/home/was/magma/packages/modform/code/modular_symbols.m°Ê	ö2¸ûÿûÿûÿûÿìËóÌËÈËœ3
÷2mfdevelø2ÿÿÿÿÿÿÿÿøò3-/usr/local/Magma2.8/package/ModFrm/newforms.mù24/usr/local/Magma2.8/package/ModFrm/modular_symbols.mË	hÌ	30/home/was/magma/packages/modform/code/newforms.ms.mË	hÌ	þ2
__sig__1¸ý2ÌËøËûÿÿÿÌŸ6þÿýÿ®®ûÿûÿûÿûÿŒÚ„ÚäÖ€Ú˜¤6Base extension of eps to R.t identi£6ýÿ¯¯ûÿûÿûÿûÿœÚ”ÚäÖÚ¦6
	__sig__52®ûÿ¥6äÖ˜Úûÿÿÿ,¼ç¨6Base extension of eps to R.”Úûÿÿÿ§6þÿýÿ¯¯ûÿûÿûÿûÿ¬Ú¤ÚäÖ Úª6
	__sig__53¯ûÿ©6äÖ¨Úûÿÿÿ,¼ç¬6#Base extension of G to Codomain(f).«6®®ûÿûÿûÿûÿ¼Ú´ÚäÖ°Ú®6
	__sig__54ûÿûÿûÿûÿ­6äÖ¸Úûÿÿÿ__si°6%Base extension of eps to Codomain(R).$¯6¯¯ûÿûÿûÿûÿHàÄÚäÖÀÚ²6
	__sig__55ûÿûÿûÿûÿ±6äÖÈÚûÿÿÿ__si´6DThe distinguished root of unity in the base ring of G and its order.extensi³6®þÿûÿûÿûÿÔÚäÖÐÚ¯ûÿ¶6
	__sig__560Ñ	ðÑ	µ6äÖØÚûÿÿÿ¸6The Kronecker character (D/n).The ·6¯ûÿûÿûÿûÿôÚäÚäÖàÚº6
	__sig__57þÿ¹6äÖèÚûÿÿÿ,¼çÀ6
	__sig__58	–	v8
tI»6DistinguishedRootØKðK¼6ÿÿÿÿÿÿÿÿÌÚ¾6ZThe Kronecker character (D/n) over the ring R, where D is a fundamental discriminant or 1.½6ýÿ¯ûÿûÿûÿûÿüÚäÖøÚ×Á6KroneckerCharactertZœZÂ6ÿÿÿÿÿÿÿÿÜÚô<auxiliary_level/modss/codeÄ6;Representatives for the Gal(Qbar/Q)-conjugacy classes of G.(–	Ã6®ûÿûÿûÿûÿÛÛäÖÛ,¼çÆ6
	__sig__59	0–	Î6
(h¢;ReducedMatrix	PÖ	X	ÜH	È6;Representatives for the Gal(Qbar/Q)-conjugacy classes of S.änŒoÇ6ÿ¯ûÿûÿûÿûÿ$ÛäÖ ÛDuÊ6
	__sig__60,vDvÉ6äÖ(ÛûÿÿÿÔwË6GaloisConjugacyRepresentativesäyÌ6ÿÿÿÿÿÿÿÿÛÍ6¯¯ûÿûÿûÿûÿ<ÛäÖ8ÛÐ6
	__sig__61Ï6äÖ@ÛûÿÿÿˆÒ61/home/was/magma/packages/modsym/code/eisenstein.mÿÿÑ61/home/was/magma/packages/modsym/code/eisenstein.m¯ûÿÔ6#ûÿûÿûÿûÿdÛ„óDÛ×ûÿÿÿÕ6MinimalBaseRingCharacter™¨™Ð™Ö6ÿÿÿÿÿÿÿÿ4Û×6//usr/local/Magma2.8/package/ModSym/eisenstein.m˜ß	à6//home/was/magma/packages/modsym/code/elliptic.m˜ß	Ú6*Compute the q-expansion of the Eisenstein series associated to the primitive character epsilon, normalized so that the coefficient of q is 1.Ù6¯#ûÿûÿûÿûÿlÛDÛhÛ__siÜ6
__sig__1`ÛÝ6PrimitiveEisensteinSeriesô°±Þ6ÿÿÿÿÿÿÿÿPÛß6//home/was/magma/packages/modsym/code/elliptic.mðÝ	â6¶VVûÿûÿûÿ”Ûˆó|Û`Û˜Ý	â<|Û„Ûûÿÿÿæ6AThe standard elliptic periods w1 and w2, computed to precision n.˜Ý	ðÝ	è6
__sig__18¿ã6-/usr/local/Magma2.8/package/ModSym/elliptic.m	˜ß	å6¶VVûÿûÿûÿœÛ|Û˜ÛüÎç6|Û Ûûÿÿÿ@ß	î6
__sig__2PÒé6
EllipticPeriods€Óí6|Û¸Ûûÿÿÿ8ÕD5./home/was/magma/packages/modsym/code/compgrp.m	ðß	H5,/usr/local/Magma2.8/package/ModSym/compgrp.mß	Xò	X5-/home/was/magma/packages/modsym/code/decomp.m	Xò	T5BdEDrderK5IThe order of the group of Fp rational points of the component group of M.pbar-pointJ5¶ûÿûÿûÿûÿ0ÕÕ,Õd toM5
__sig__1ry that p eL5Õ4ÕûÿÿÿequaY5__sig__0ponentGroupOfJ0N rQ5 fCompute the order of the image of the component group of J_0(N) in the component group of A, all at p.	Èâ	R5ÕLÕûÿÿÿŠ6
ctorsnN5ComponentGroupOrderP$h$O5ÿÿÿÿÿÿÿÿÕP5¶ûÿûÿûÿûÿHÕÕDÕì,W5-/home/was/magma/packages/modsym/code/decomp.m	`ä	Z5¶ûÿûÿûÿûÿ4Útó\Õ$ÕûÿÿÿU5¡\¢V5Mä:ü:‚5__sig__7ac ä	Ðä	ÈS	ä»çI5"kDecomposition of M with respect to the Hecke operators T_p with p coprime to the level of M and p<= bound. Ðä	¸å	Ý<\ÕdÕûÿÿÿr5
__sig__4pace$Õb5
__sig__1_sig__2Fa5\ÕˆÕûÿÿÿ´I[5!OrderOfImageOfComponentGroupOfJ0N0MPM\5ÿÿÿÿÿÿÿÿ@Õ]5+/usr/local/Magma2.8/package/ModSym/decomp.mæ	Xò	š5+/home/was/magma/packages/modsym/code/dims.mØæ	Xò	`54¯Unsorted decomposition of M into factors corresponding to the Galois conjugacy classes of newforms of level a divisor of the level of M. We require that IsCuspidal(M) is true.ûÿ_5¶ûÿûÿûÿûÿ„Õ\Õ€Õûÿÿÿd5DóThe space of modular symbols corresponding to the Galois-conjugacy class of newforms associated to M. The level of the newforms is allowed to be a proper divisor of the level of M. The space M must have been created using NewformDecomposition.c5¶¶ûÿûÿûÿûÿ”Õ\ÕÕøé	f5
__sig__2Ìtg5NewformDecompositionuvh5ÿÿÿÿÿÿÿÿ|Õj5ATrue if and only if M was constructed using NewFormDecomposition. class of i5¶ûÿûÿûÿûÿ¬Õ\Õ¨Õf M.l5
__sig__3reated usink5\Õ°Õûÿÿÿp58¸Sort the sequence of spaces of modular symbols with respect to the 'lt' comparison operator. Each space must corresponding to a single Galois-conjugacy class of newforms of some level.ÿÿÿÈ’s5HasAssociatedNewSpace8”t5ÿÿÿÿÿÿÿÿ¤Õv5FûDecomposition of subspace of M obtained using T_2 - a_2, T_3 - a_3, ..., where a_p varies over all rational integers with |a_p| <= 2*Sqrt(p), and p varies over primes not dividing the level of M. These factors frequently correspond to elliptic curves.u5¶ûÿûÿûÿûÿÜÕ\ÕØÕxÕz5PThe kernel of I on M. This is the subspace of M obtained by intersecting the kernels of the operators f_n(T_{p_n}), where I is a sequence [<p_1, f_1(x)>,...,<p_n,f_n(x)>] of pairs consisting of a prime number and a polynomial. Only primes p_i which do not divide the level of M are used.ðï	‘	y5¶ÿ¶ûÿûÿûÿûÿ¬öìÕ\ÕèÕ,¼ç|5
__sig__6ð	‘	6
uspForms}5
EllipticFactors¼»~5ÿÿÿÿÿÿÿÿÔÕ€5R.The space M of modular symbols associated to the elliptic curve E. The existence of M is guaranteed by the Shimura-Taniyama conjecture, which has been proved by Breuil, Conrad, Diamond, Taylor, and Wiles. WARNING: Finding M is computationally difficult, even when Conductor(E) is on the order of 10000.	Xò	†5
__sig__8ÿ…5\ÕÖûÿÿÿxÕüÕ@ÓÆ4
WindingElementlÈÇ4ÿÿÿÿÿÿÿÿðÒË4
	__sig__17g element Ê4¨Ñ,Óûÿÿÿxó	Í4[The image under RationalMapping(M) of the ith winding element X^(i-1)*Y^(k-2-(i-1))*{0,oo}.	__siÏ4
	__sig__18¼ÑØ4¨ÑdÓûÿÿÿ__si5“9 \4GThe number of real components of the complex torus A_M associated to M.õ	„Œ	[4¶ûÿûÿûÿûÿtÑDópÑ as ^4
	__sig__10ô	põ	]4DóxÑûÿÿÿa4//home/was/magma/packages/modsym/code/boundary.mÈõ	Ö<
__sig__0ackages/mod_4MinusTamagawaNumbernþ];`4ÿÿÿÿÿÿÿÿ\Ñ,4//home/was/magma/packages/modsym/code/boundary.mus Ad4$pA matrix that represents the boundary map from M to the vector space whose basis consists of the weight-k cusps.z ö	`÷	×<°ÐXóûÿÿÿ5
modulenge4RealTamagawaNumberf4ÿÿÿÿÿÿÿÿlÑg4*/usr/local/Magma2.8/package/ModSym/arith.mÐ÷		h4-/usr/local/Magma2.8/package/ModSym/boundary.m		k4+/home/was/magma/packages/modsym/code/calc.m`ø	øù	j40+/home/was/magma/packages/modsym/code/calc.m¨ø	øù	i4.žThe Hecke algebra associated to M. If the optional integer parameter Bound is set, then the algebra generated only by those T_n, with n <= Bound, is computed.	øù	l4
__sig__0’`; ù	øù	y4
__sig__2keAlgebran4)/usr/local/Magma2.8/package/ModSym/calc.m];øù		q48ÆThe discriminant of the Hecke algebra associated to M. If the optional parameter Bound is set, then the discriminant of the algebra generated by only those T_n, with n <= Bound, is computed instead.p4¶ûÿûÿûÿûÿÈÑ¨ÑÄÑs4__sig__1ÿÿÿt4
HeckeAlgebrau4ÿÿÿÿÿÿÿÿ´Ñ4
__sig__3keAlgebraw4$qThe following upper bound on the torsion subgroup of A: gcd { #A(F_p) : 3 <= p <= maxp, p not dividing Level(A)}.nt of the v4¶ûÿûÿûÿûÿàÑ¨ÑÜÑx4¨ÑäÑûÿÿÿûÿ|4·¶ûÿûÿûÿûÿÒøÑ¨ÑôÑûÿÿÿ5\ÕÖûÿÿÿûÿÿÿÈ5Orders¤ˆ5&zThe 0, +1 or -1 quotient of the space M of modular symbols over K cut out by the system of eigenvalues corresponding to E. conjectu‡5õÿ&¶ûÿûÿûÿûÿ,Ö$Ö\Õ ÖtatiŠ5
__sig__9onductor(E)‰5\Õ(ÖûÿÿÿŒ5’(The space of modular symbols "[Level]k[Weight][IsogenyClass]" with trivial character and given sign corresponding to a Galois-conjugacy class of newforms. Here [Level] is the level, [Weight] is the weight, [IsogenyClass] is a letter such as "A", "B", ..., "Z", "AA", "BB", etc., and k is a place holder to separate the level and weight. If argument 1 is "[Level][IsogenyClass]", then the weight k is assumed equal to 2. If argument 1 is "[Level]k[Weight]" then the cuspidal subspace of the full ambient space of modular symbols of weight k is returned.&‹5¶ûÿûÿûÿûÿ<Ö4Ö\Õ0Ö°	Ž5
	__sig__10¸05\Õ8Öûÿÿÿ of 5,The full space of modular symbols "Nk[Weight][IsogenyClass]" with trivial character and corresponding to a Galois-conjugacy class of newforms. ¶ûÿ5¶ûÿûÿûÿûÿPÜDÖ\Õ@Öûÿÿÿ’5
	__sig__11(The ‘5\ÕHÖûÿÿÿsoge”51/home/was/magma/packages/modsym/code/derivative.mois-conjug“51/home/was/magma/packages/modsym/code/derivative.mht, [Isoge^5$sThe leading coefficient of Taylor expansion about the critical point 1 and the order of vanishing of L(M,s) at s=1. 1 is "[–5¶VûÿûÿûÿxóLÖxÕthen•5
__sig__0 full ambieÞ<LÖTÖûÿÿÿetur¯5dÖÀÖûÿÿÿµ5dÖØÖûÿÿÿûÿà<äÖìÖûÿÿÿ__siÍ5®ûÿûÿûÿûÿL×<×äÖ8×f moÛ5äÖp×ûÿÿÿh trÝ5®“ûÿûÿûÿ|×äÖx×à5
__sig__4ß5äÖ€×ûÿÿÿœlâ5The Dirichlet characters in G.	ê5
(p—5//usr/local/Magma2.8/package/ModSym/derivative.m	™5
+/home/was/magma/packages/modsym/code/dims.me.m	»5
__sig__0pGamma1˜5OCompute the dimension of the space of cusp forms of weight k and character eps.rder of vaniœ5¯þÿûÿûÿûÿûÿ|ódÖ`ÖV›5
__sig__0…È		ˆ
	ß<dÖlÖûÿÿÿb6
tRing,‹5LSeriesLeadingCoefficientˆŽÀŽž5ÿÿÿÿÿÿÿÿXÖŸ5)/usr/local/Magma2.8/package/ModSym/dims.m		 5*[The values [eps(1), eps(2), ..., eps(N)].H		¢5!The dimension of S_k(Gamma_0(N)). of cusp f¡5ûÿûÿûÿûÿŒÖdÖˆÖ¤5
__sig__1Ö€Ö£5dÖÖûÿÿÿ|¢¥5DimensionCuspForms¤¤¼¤¦5ÿÿÿÿÿÿÿÿpÖÂ5
RootOf1ormsGamma1¨5!The dimension of S_k(Gamma_1(N)).«\«§5ûÿûÿûÿûÿ¤ÖdÖ Ö¯ª5
__sig__2°p°˜°©5dÖ¨Öûÿÿÿ²«5DimensionCuspFormsGamma0³ø³´¬5ÿÿÿÿÿÿÿÿ„Ö®55The dimension of the new subspace of S_k(Gamma_0(N)).	­5ûÿûÿûÿûÿ¼ÖdÖ¸Ö,¼ç°5
__sig__3	Ø‘	6
Ì¾±5DimensionCuspFormsGamma1À¤À¼Á²5ÿÿÿÿÿÿÿÿœÖ´5=Compute the dimension of the new subspace of S_k(Gamma_0(N)).	Ø	³5ûÿûÿûÿûÿÔÖdÖÐÖ	ü5
uppÒz4DiscriminantOfHeckeAlgebra yèy{4ÿÿÿÿÿÿÿÿÀÑ}4&zThe cyclic subgroup of the complex torus attached to M that is generated by the image under the period map of the point x.x	p	~4¨ÑüÑûÿÿÿ40¦An abelian group that is isomorphic to the finite group generated by the sequence of images pi(s[i]) in the complex torus attached to M, where pi is PeriodMapping(M).d by €4ÿ·¶“ûÿûÿûÿûÿÒ¨ÑÒƒ4
__sig__4ÿÒ‚4¨ÑÒûÿÿÿ›ˆ4¨Ñ$Òûÿÿÿä„4
SubgroupOfTorus\Ÿ…4ÿÿÿÿÿÿÿÿðÑ‡4<ÏThe kernel of the modular isogeny. Let T be the complex torus attached to M. Then the modular isogeny is the natural map from the dual of T into T induced by autoduality of CuspidalSubspace(AmbientSpace(M)).¶†4¶“ûÿûÿûÿûÿ Ò¨ÑÒà	‰4
__sig__5±‹4&}This is Sqrt(#ModularKernel(M)). If A_M is an elliptic curve E, then this degree of the map of algebraic curves X_0(N) --> E.the duŠ4¶ûÿûÿûÿûÿ0Ò¨Ñ,Ò4
__sig__6Œ4¨Ñ4Òûÿÿÿ°¼Ÿ4
__sig__9½½¨½ž4¨Ñ|Òûÿÿÿl¿Ž4
ModularKernel\Ã4ÿÿÿÿÿÿÿÿÒ–4¶ûÿûÿûÿûÿ`Ò¨Ñ\Ò of ™4
__sig__8_0(N) --> E˜4¨ÑdÒûÿÿÿ@Ñ07hc²6³´Šµ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞŒßàáâãäåæçèéêëìíîïðñòóô|õz~ö÷øùúûüýþÿ	
®7-The weight of the space M of modular symbols.haract­7¶ûÿûÿûÿûÿ¼ÞœÜ¸Þwork°7
__sig__8 possible v¯7œÜÀÞûÿÿÿthe ²7,The level of the space M of modular symbols.¸Ü±7¶ûÿûÿûÿûÿøçÌÞœÜÈÞp	´7
__sig__9³7œÜÐÞûÿÿÿ¸Ü¶7QTrue if and only if M is contained in the cuspidal subspace of the ambient space.	ì™	µ7¶ûÿûÿûÿûÿÜÞœÜØÞ¸7
	__sig__10è·7œÜàÞûÿÿÿ!º7STrue if and only if M is contained in the Eisenstein subspace of the ambient space.tained i¹7¶ûÿûÿûÿûÿìÞœÜèÞ¼7
	__sig__11»7œÜðÞûÿÿÿ0¾70The base ring of the space M of modular symbols.STrue½7¶ýÿûÿûÿûÿûÿ8÷üÞœÜøÞÀ7
	__sig__12¿7œÜßûÿÿÿ¼^Â71The base field of the space M of modular symbols.ÿÿÁ7¶õÿûÿûÿûÿûÿßœÜßÄ7
	__sig__13Ã7œÜßûÿÿÿ¸vÆ7:The Dirichlet character of the space M of modular symbols.´ƒÅ7¶ûÿûÿûÿûÿßœÜßÈ7
	__sig__14Ç7œÜ ßûÿÿÿP"	Ê7(True if and only if the sign of M is +1.ÿÿÿÉ7¶ûÿûÿûÿûÿ,ßœÜ(ß#	Ì7
	__sig__15Ë7œÜ0ßûÿÿÿ¸ÜÎ7(True if and only if the sign of M is -1.¸ÜÍ7¶ûÿûÿûÿûÿ<ßœÜ8ß$	Ð7
	__sig__16Ï7œÜ@ßûÿÿÿ¸ÜÔ7"The sign of M; either -1, 0, or 1.	Xš	`8
˜ÉÑ7
IsPlusQuotient¬ÉÒ7ÿÿÿÿÿÿÿÿ$ßÓ7¶ûÿûÿûÿûÿTßœÜPßÖ7
	__sig__17ûÿÕ7œÜXßûÿÿÿ,¼çÚ7The dimension of M.¸Ü×7
IsMinusQuotient×Ø7ÿÿÿÿÿÿÿÿ4ßÙ7¶ûÿûÿûÿûÿÄõlßœÜhßÜ7
	__sig__18ûÿûÿûÿûÿÛ7œÜpßûÿÿÿ__siä7
	__sig__20rus¸ÜÞ73The dimension of the abelian variety attached to A.ÈÉG7
plus_part_sub¨H7
plus_part_dualªI7
minus_part_sub\«J7minus_part_dualø¬­8­K7
plus_subspace˜®L7
minus_subspace°M7
new_part±(±P±N7
old_part²€²˜²O7

p_new_partø³´P7
modsym_basis´µQ7associated_new_space¶h¶R7star_involutionX¸p¸ˆ¸S7dual_star_involutionº¨ºT7
discriminant»ð»U7
discriminant_Qà¼V7
winding¨½ð½¾W7twisted_windingl¿„¿Ô¿X7twisted_winding_index„ÄY7
boundary_mapÅ´ÅZ7
cusplistÈXÈ`Ê[7
atkin_lehnerÌHÌ\7dual_atkin_lehnerHÎ Î]7dual_hecke_operator(+	8,	^7standard_imagesÔÑìÑÒ_7standard_images_alllÓ„Ó`7degeneracy_matrices_out<ÕpÕ Õa7degeneracy_matrices_inÈ×U6¯ûÿûÿûÿûÿ\ÙäÖXÙûÿX6
	__sig__33	,¼çW6äÖ`Ùûÿÿÿ×Z6(…True if and only if Evaluate(x,-1) is equal to 1. Note that the space of modular forms of weight k is zero if x is even and k is odd.ûÿY6¯ûÿûÿûÿûÿlÙäÖhÙûÿÿÿd6
	__sig__36…Truec6äÖÙûÿÿÿ. Non6äÖ´Ùûÿÿÿ k ip6¯¯¯ûÿûÿûÿûÿ´áÈÙäÖ¼Ùq6
	__sig__41tÙr6äÖÄÙûÿÿÿX/	x6<ÐThe quotient of x and y. This is a Dirichlet character of modulus equal to the least common multiple of the moduli of x and y. The base rings and chosen roots of unity of the parents of x and y must be equal.+@.	H0	z6
	__sig__42er-.e6¯þÿûÿûÿûÿûÿÌÙœÙäÖ˜Ù¯ûÿh6
	__sig__37h0	(1	g6äÖ Ùûÿÿÿ|6
 :i6AssociatedPrimitiveCharacter<0<j6ÿÿÿÿÿÿÿÿ„Ùk6The trace of f. x at n.l6oþÿûÿûÿûÿûÿ¸ÙäÖ¬Ù×m6
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“¢7T0The space of modular symbols of level N, weight k, trivial character, and given sign over the rational numbers. If sign=+1 then returns the +1 quotient, if sign=-1 returns the -1 quotient, and if sign=0 returns the full space, respectively. The +1 quotient of M is M/(*-1)M, where * is StarInvolution(M).Ü<Þ´¤7
__sig__5ƒ¬ƒ£7œÜÞûÿÿÿÜ‡¦7;The space of modular symbols of weight k and character eps.haracter¥7¯¶ûÿûÿûÿûÿ¤ÞœÞœÜ˜Þ¨7
__sig__6ÞÞ¸Ü§7œÜ Þûÿÿÿª7NThe space of modular symbols of weight k and character eps. The level and base field are specified as part of eps. The third argument "sign" allows for working in certain quotients. The possible values are -1, 0, or +1, which correspond to the -1 quotient, full space, and +1 quotient.re * i©7¯¶ûÿûÿûÿûÿÌö¬ÞœÜ¨Þ¶ûÿ¬7
__sig__7ÃpJ	0K	«7œÜ°Þûÿÿÿç7¶ûÿûÿûÿûÿ\ô¤ßœÜ ßractì76The ambient space of modular symbols, in which M lies.ë7¶¶ûÿûÿûÿûÿlô´ßœÜ°ß¸Üî7
	__sig__22	Ð™	í7œÜ¸ßûÿÿÿò7
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	__sig__24D›Ð5__sig__1ÿÿÿôœÒ5ÿÿÿÿÿÿÿÿðÖÔ5AThe group of Dirichlet characters mod m with image in the ring R.d(). This Ó5ýÿ®ûÿûÿûÿûÿd×T×äÖP×Ö5
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	__sig__34€××[6äÖpÙûÿÿÿ`Ê]6¯ûÿûÿûÿûÿ|ÙäÖxÙ`6
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__sig__3pace.*\ß8$ã€ãûÿÿÿä8&qThe "even modular symbol phi_M^+", as on page 209 of Mazur, "On the Arithmetic of Special Values of L-functions."ÿÿìá8DualModularSymbol$Dâ8ÿÿÿÿÿÿÿÿ\ãã8¶uûÿûÿûÿûÿ”ã$ããf Spæ8
__sig__4""å8$ã˜ãûÿÿÿê8$pThe "odd modular symbol phi_M^-", as on page 209 of Mazur, "On the Arithmetic of Special Values of L-functions."ì*ç8ModularSymbolEven-Œ-è8ÿÿÿÿÿÿÿÿtãé8¶uûÿûÿûÿûÿ´ã¬ã$ã¨ãf Spì8
__sig__5Ä9Ü9ë8$ã°ãûÿÿÿî8$pThe "odd modular symbol phi_M^-", as on page 209 of Mazur, "On the Arithmetic of Special Values of L-functions."¤?í8¶uûÿûÿûÿûÿ¼ã$ã¸ã Speð8
__sig__6EüEFï8$ãÀãûÿÿÿô81/home/was/magma/packages/modsym/code/qexpansion.mç__siõ8
__sig__0@ãñ8ModularSymbolOddPQ0Qò8ÿÿÿÿÿÿÿÿ¤ãó81/home/was/magma/packages/modsym/code/qexpansion.mf Mazur, "Ð8<ÕThe q-expansion of one of the Galois-conjugate newforms associated to M, computed to absolute precision prec. The coefficients of the q-expansion lie in a quotient of a polynomial extension of the base field of M.ûÿÿÿ÷8//usr/local/Magma2.8/package/ModSym/qexpansion.mg	$9For internal use.odsym/codeú8@äThe integral basis of q-expansions for the sum of the spaces of modular forms associated to the modular symbols spaces in D, computed to absolute precision prec. The base field must be either the rationals or a cyclotomic field.Ÿ	ù8ûÿûÿûÿûÿ$äìãÌãèã,¼çü8
__sig__1^	Ÿ	Û9
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__sig__3 The base f9Ìã äûÿÿÿtomi	9¶ûÿûÿûÿûÿ,äÌã(ä8ä9
__sig__4¼ç__si9Ìã0äûÿÿÿûÿÿÿ9
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__sig__7tion of M ið<Ìã¼óûÿÿÿutin>9
__sig__6n taking al=9läøäûÿÿÿ If \9
__sig__4" then the [9åpåûÿÿÿsal ^90¢The new subspace of M. This is the intersection of NewSubspace(M,p) as p varies over all prime divisors of the level of M. Note that M is required to be cuspidal.erators.]9¶¶ûÿûÿûÿûÿ|ååxåàã`9
__sig__5_sig__3Õ_9å€åûÿÿÿd9
__sig__6_sig__4Ûc9ååûÿÿÿ”ßÝ7¶ûÿûÿûÿûÿ|ßœÜxß,à7
	__sig__19@ß7œÜ€ßûÿÿÿXê7
	__sig__21ruslâ79The degree of M, which is the dimension of the root of M.xh	0i	á7¶ûÿûÿûÿûÿŒßœÜˆßˆi	ã7œÜßûÿÿÿ\8
ntModSymå7DimensionComplexTorus¤æ7ÿÿÿÿÿÿÿÿtßè7$vTrue if and only if M is the ambient space of modular symbols, which was created by specifying a weight and character.é7œÜ¨ßûÿÿÿÀ	ð7(„True if and only if the Hecke operators T_p, with p prime to the level of M, do not decompose M into smaller modular symbols spaces.pecifyiï7¶ûÿûÿûÿûÿÄßœÜÀßûÿñ7œÜÈßûÿÿÿ,¼çô7UTrue if and only if M is contained in the new cuspidal subspace of the ambient space.ace ofó7¶ûÿûÿûÿûÿÜßÔßœÜÐßõ7œÜØßûÿÿÿHm	ø7WTrue if and only if M is contained in the p-new cuspidal subspace of the ambient space.and only if ÷7¶ûÿûÿûÿûÿäßœÜàßbolsú7
	__sig__25ù7œÜèßûÿÿÿûÿü7The basis of M.¨n	,¼çû7¶ûÿûÿûÿûÿŒôôßœÜðßand þ7
	__sig__26new cuspidý7œÜøßûÿÿÿ€o	8:ÍA basis over the integers for the integral modular symbols. This is the intersection of M with the Z-lattice generated by all modular symbols X^iY^(k-2-i){a,b}. The base field of M must be RationalField().is conÿ7¶ûÿûÿûÿûÿàœÜà8
	__sig__27ûÿ8œÜàûÿÿÿ,¼ç8Coerce x into M.¸Ü8¶·ûÿûÿûÿàœÜà8
	__sig__28ûÿ8œÜàûÿÿÿ,¼ç8HThe modular symbols subspace of M1 associated to M2. Let N1 be the level of M1. If ModularSymbols(M2,N1) is defined, let M3 be this modular symbols space, otherwise terminate with an error. If M3 is contained in M1, return M3, otherwise terminate with an error.ationa8¶¶¶ûÿûÿûÿûÿ$àœÜ à
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	__sig__29ç	__si	8œÜ(àûÿÿÿûÿÿÿ8Base extension of M to F.Manin symb8õÿ¶¶ûÿûÿûÿûÿ¤áPàœÜDà8
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`œ'7l-/home/was/magma/packages/modsym/code/modsym.m	u	@8
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hecke_boundœ­´­37is_ambient_spacez	Ô¿ç67mlist³57
other_levels²T²87Nmension77quot´µ;7dual_representationµ¶97F¼¸Ô¸:7sub_representation¼ºÔº=7dual_cuspidal_representation¸<7sub_cuspidal_representationð¾¿?7dual_integral_representation¸g7mestre¿>7sub_integral_representation´Ç4ÈA7%dual_integral_cuspidal_representationf7XÀÁ@7$sub_integral_cuspidal_representationÐìÐE7
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complement,ÛTÛtÛ¤Ûê6ÿÿÿÿÿÿÿÿˆÛë6¶VVVûÿûÿ¼Û´Û|Û°Û|	ð66The invariants c4, c6, and j, computed to precision n.ï6¶VVVûÿûÿÄÛ|ÛÀÛ¸|	ò6
__sig__3ñ6|ÛÈÛûÿÿÿÛø6
__sig__4_sig__2÷6|ÛàÛûÿÿÿ<ó6EllipticInvariantst”ô6ÿÿÿÿÿÿÿÿ¬Ûö6"nAn elliptic curve over the rational numbers that lies in the isogeny class of elliptic curves associated to M.õ6¶ûÿûÿûÿûÿÜÛ|ÛØÛÿ6
__sig__0SmallPrimeú6]The space of mod ell modular symbols associated to E, where ell should be a very small prime.ed to ù6&¶ûÿûÿûÿûÿìÛ|ÛèÛÛü6
__sig__5_sig__46û6|ÛðÛûÿÿÿ9ä<
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MatrixGrouppackage/‹80á0âûÿÿÿ__si%7decomp­79/usr/local/Magma2.8/package/ModSym/intersection_pairing.m@‚	XŽ	7-/home/was/magma/packages/modsym/code/linalg.m	èƒ	7+/home/was/magma/packages/modsym/code/maps.mà‚	èƒ	7+/home/was/magma/packages/modsym/code/maps.m(ƒ	èƒ	7¶¶ûÿûÿûÿûÿ”óÜ Ü	7IntersectionPairingøde
7ÿÿÿÿÿÿÿÿÜ7+/usr/local/Magma2.8/package/ModSym/linalg.mX„	XŽ	7)/usr/local/Magma2.8/package/ModSym/maps.mm „	XŽ	7-/home/was/magma/packages/modsym/code/mestre.m	XŽ	7(€The matrix of DegeneracyMap(M1,M2,d) with respect to Basis(M1) and Basis(M2). Both IsAmbient(M1) and IsAmbient(M2) must be true.numerator o7¶¶ûÿûÿûÿûÿ@ÜÜ<Ü int7
__sig__1---> M_2, o7ÜDÜûÿÿÿturn7¶¶ûÿûÿûÿûÿ<ÞXÜÜTÜ7
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__sig__0repsgene7*The supersingular j-invariants in GF(p^2).IsAmbient7ûÿûÿûÿûÿ˜óhÜ4Ü7
__sig__0\Ü4Üæ<hÜpÜûÿÿÿx¢ç<œÜ¤Üûÿÿÿ¸¤7DegeneracyMatrix¦H¦p¦7ÿÿÿÿÿÿÿÿ8Ü7:ÉAmbient modular symbols spaces M can create circular references, which cause the memory manager to never de-allocate M. Calling DisownChildren forces the circular references created by M to be deleted.ntrinsic r7 cThe space of modular symbols of level N, weight 2, and trivial character over the rational numbers.t divide7¶ûÿûÿûÿûÿTÞDÞœÜ@Þ‘7œÜHÞûÿÿÿ(Œ	˜7
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__sig__1Ü¤Üûÿÿÿ‡6ExtendHËˆ6ÿÿÿÿÿÿÿÿüÙ‰6¯ûÿûÿûÿûÿ”á,ÚäÖ(ÚŒ6
	__sig__46ûÿûÿûÿûÿ‹6äÖ0Úûÿÿÿ__si8/An expression of x in terms of modular symbols.ûÿÿÿ M2.8œÜ€àûÿÿÿs(M2D8···ûÿûÿûÿûÿ ááœÜáontaE8
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__sig__9SL2Ìå¹9The element of G defined by S. a ra¸9¬­ûÿûÿûÿèæ÷äæØ§	»9
__sig__8¬º9÷ìæûÿÿÿÌå¾9÷üæûÿÿÿp!Ë9
	__sig__112EltÄ9÷çûÿÿÿà$¼9¬ûÿûÿûÿûÿûÿøæ÷ôæÂ9­ûÿûÿûÿûÿûÿç÷çP©	Å9
	__sig__10 KÀ9HackobjCoerceGrpPSL2¸KÁ9ÿÿÿÿÿÿÿÿàæÊ9÷,çûÿÿÿ×9
	__sig__13L2EltÎ9­¬ûÿûÿûÿûÿ@ç÷<çÌ^Æ9HackobjPrintGrpPSL2ä^Ç9ÿÿÿÿÿÿÿÿðæÝ9
	__sig__14L2Elt(gÉ9Returns true if A is in G.È9¬ûÿûÿûÿûÿ(ç÷$çÕ9
Ð‡Ì9HackobjPrintGrpPSL2EltHŠÍ9ÿÿÿÿÿÿÿÿçÑ9
	__sig__12ue if A isÐ9÷DçûÿÿÿÔ9­­ûÿûÿûÿûÿhçXç÷Tç3Ö9÷\çûÿÿÿø²Ò9HackobjInGrpPSL2·h·Ó9ÿÿÿÿÿÿÿÿ ç3:
ceX»ß96Returns true if A is equivalent to B under action of GÿûÿÞ9¬­­ûÿûÿûÿûÿˆç€ç÷|çÐÃØ9HackobjParentGrpPSL2EltìÃÙ9ÿÿÿÿÿÿÿÿ8çÚ9¬¬ûÿûÿûÿûÿôêpç÷lçÐÉÜ9÷tçûÿÿÿÀ¯	á9
	__sig__15üÉà9÷„çûÿÿÿã96Returns true if A is equivalent to B under action of GÌåå9
	__sig__16sig__16ä9÷”çûÿÿÿ$× :IsGammasâ9¬kkûÿûÿûÿûÿÀìç÷Œçç9"tests whether G is a subgroup of Hœçæ9¬¬ûÿûÿûÿûÿ ç÷œç8Ýé9
	__sig__17ue if A isè9÷¤çûÿÿÿˆ²	ë9$tests whether G is finite index in Hûÿê9¬¬ûÿûÿûÿûÿ°ç÷¬çûÿÿÿí9
	__sig__18"testì9÷´çûÿÿÿˆ³	ñ9*returns index of G in H, if this is finiteÿûÿûÿûÿð9¬¬ûÿûÿûÿûÿØçÈç÷Äç28œÜÄàûÿÿÿØ_8¶ûÿûÿûÿûÿûÿ„áœÜ€áb8
	__sig__31ûÿa8œÜˆáûÿÿÿ,¼ç 8
	__sig__33Elt	8
d8ÿÿÿÿÿÿÿÿ8àe8·ûÿûÿûÿûÿ˜éœáœÜ˜áh8
	__sig__51¸ág8œÜ áûÿÿÿ0j8HBase extension of M to Codomain(f) using the map f : BaseField(M) --> F.i8¶¶ûÿûÿûÿûÿ¬áœÜ¨á·	l8
	__sig__52´ y8···ûÿûÿûÿûÿôáðáœÜèál$L80/home/was/magma/packages/modsym/code/operators.mûÿx·	Øº	M8FA matrix representing the nth Hecke operator with respect to Basis(M).ûÿÿÿN8¶ûÿûÿûÿûÿö ó0á4á‚8
__sig__1áÐá¸Ü80áâûÿÿÿ 9‰8·Ôûÿûÿûÿûÿ,â0á(â•8¶ûÿûÿûÿûÿûÿ\â0áXâ__si˜8
__sig__5¸Ü—80á`âûÿÿÿ
›8¶ûÿûÿûÿûÿtâ0ápâ¸Ü80áxâûÿÿÿxJ¤8__sig__7ÿÿÿŒM8œÜâûÿÿÿK80/home/was/magma/packages/modsym/code/operators.mûÿ»	˜»	è<0áìáûÿÿÿ__siG8
ÈU†8
__sig__2atorsentinQ8XA matrix representating the nth Hecke operator with respect to the basis for Lattice(M).R8¶ûÿûÿûÿûÿâ0áDá<á„8^Computes the Hecke module over Z generated by a vector v. The result is returned as a lattice.s forƒ8·Ôûÿûÿûÿûÿâ0áâûÿ…80áâûÿÿÿ__siŒ8
__sig__3<á‡8IntegralHeckeOperatorèpˆ8ÿÿÿÿÿÿÿÿHáŠ8.Computes the Hecke module over the base field generated by a vector v. The result is returned as a subspace of the vector space underlying the ambient space.’8
__sig__4á$âh}‘80áHâûÿÿÿ88
	HeckeSpan°€È€Ž8ÿÿÿÿÿÿÿÿâx9Level8¶ûÿûÿûÿûÿDâ0á@ât is–8<ÓMany computations require a bound n such that T1,...,Tn generate the Hecke algebra as a Z-module. This command allows you to set the bound that is used internally. Setting it too low can result in false answers.`	û<““8
HeckeFieldSpan´œ”8ÿÿÿÿÿÿÿÿ$â¨8zÊThe matrix that represents the qth Atkin-Lehner involution W_q on M, when it is defined on modular symbols. The involution W_q is defined on modular symbols when M has trivial or quadratic character and even weight (otherwise it doesn't preserve M). When possible, the Atkin-Lehner operator is normalized so that it is an involution; such normalization may not be possible when the weight k of M is >2 and the characteristic of the base field of M divides q.§8¶ûÿûÿûÿûÿ\ö¤â0á â__si©80á¨âûÿÿÿûÿÿÿ™8

HeckeBound¹P¹š8ÿÿÿÿÿÿÿÿ<âœ8`bCompute a matrix representing the Hecke operator T_n on the dual representation of M. This function is takes significantly less time to run than HeckeOperator(M,n) when the dimension of M is small relative to the dimension of the ambient space containing M. Note that DualHeckeOperator(M,n) is not guaranteed to equal the transpose of HeckeOperator(M,n).Ï¨Ïž8__sig__6ÿÿÿ€ÑŸ8
SetHeckeBoundÔÒ 8ÿÿÿÿÿÿÿÿTâ¢8%The same as AtkinLehnerOperator(M,q).£80áâûÿÿÿ Ç	ª8
__sig__8ÜPÞHß“7
DisownChildrenø¬”7ÿÿÿÿÿÿÿÿ¨Ü–7 dThe space of modular symbols of level N, weight k, and trivial character, over the rational numbers.³°³•7¶ûÿûÿûÿûÿdÞ\ÞœÜXÞh¶—7œÜ`ÞûÿÿÿHÉ	š7[The space of modular symbols of level N, weight k, and trivial character, over the field F.nd trivi™7õÿ¶ûÿûÿûÿûÿtÞlÞœÜhÞ›7œÜpÞûÿÿÿÄÄž7"gThe space of modular symbols of level N, weight k, trivial character, and given sign, over the field F.ular symbols7õÿ¶ûÿûÿûÿûÿ„Þ|ÞœÜxÞ 7
__sig__4¶ûÿŸ7œÜ€Þûÿÿÿ,¼ç¡7¶ûÿûÿûÿûÿ”ÞŒÞœÜˆÞÙ ÚhÚK9-/usr/local/Magma2.8/package/ModSym/subspace.m	à	j9./home/was/magma/packages/modsym/code/verbose.m	à	P9
__sig__1subspace ofO9å@åûÿÿÿR9WThe Eisenstein subspace of M. The is the complement in M of the cuspidal subspace of M.Í	pÍ	Q9¶¶ûÿûÿûÿûÿ|öLååHåkernT9
__sig__2ÐÌ	8Î	S9åPåûÿÿÿV9A synonym for Complement(M).Í	Î	U9¶¶ûÿûÿûÿûÿèè\ååXåThe X9
__sig__3 is the comW9å`åûÿÿÿZ9V>The p-new subspace of M. This is the complement in M of the subspace generated by the modular symbols of level equal to the level of M divided by p and character the restriction of the character of M. If the character of M does not restrict, then NewSubspace(M,p) is equal to M. Note that M is required to be cuspidal.üèY9¶¶ûÿûÿûÿûÿtålååhå6b9,’The space of modular symbols complementary to M in AmbientSpace(M). Thus the ambient space of M is equal to the direct sum of M and Complement(M). and chara9¶¶ûÿûÿûÿûÿŒååˆåce(Mf9$tThe old subspace of M. This is simply the complement in M of NewSubspace(M). Note that M is required to be cuspidal.ÿûÿûÿe9¶¶ûÿûÿûÿûÿœåå˜åh9
__sig__7he new subsg9å åûÿÿÿbspal9+/home/was/magma/packages/modss/code/modss.m.m	×	L9
ModularSymbolsTû$èÐÔø'ðCTDÄvx,xxxˆyÌyP} pH…Ô…PŒäŒP4 HH‘”Ã¨ÐÄå0ñLÐ0å÷ü$ó<\/0‰12Ý°{34†o5=
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ç¶ûÿý<´ûÿûÿûÿûÿôüó¬åðóûÿÿÿþ<
__sig__0tThe ÿ<¬åøóûÿÿÿment=SupersingularModulerequired=ÿÿÿÿÿÿÿÿôó=
¶ûÿ=´ûÿûÿûÿûÿô¬åôûÿÿÿm9
OldSubspaceÈzàzn9ÿÿÿÿÿÿÿÿ”åo9,/usr/local/Magma2.8/package/ModSym/verbose.m×	à	q9
ModularSymbols$‰p90/usr/local/Magma2.8/package/GrpPSL2/GrpPSL2.spec(\¨;
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mMatrix	r9:/usr/local/Magma2.8/package/GrpPSL2/congruence_subgroups.mÐØ	à	s9:/usr/local/Magma2.8/package/GrpPSL2/congruence_subgroups.m(Ù	à	w9MatrixRepresentation›œ“9ÿÿÿÿÿÿÿÿ æ•9*creates the congruence subgroup Gamma_0(N)nce subgr”9¬ûÿûÿûÿûÿXæ÷Tæd n,—9
__sig__3irichlet ch–9÷\æûÿÿÿ™9*creates the congruence subgroup Gamma_1(N)\æÌå˜9¬ûÿûÿûÿûÿhæ÷dæèÛ	›9
__sig__4uence subgrš9÷læûÿÿÿŸ9&yThe congruence subgroup Gamma_0(N), Gamma_1(N), Gamma(N), Gamma^1(N), or Gamma^0(N), when k = 0,1,2,3, or 4 respectively.hæûÿÿÿ9ÿÿÿÿÿÿÿÿPæž9¬ûÿûÿûÿûÿÀæ€æ÷|æ¡9
__sig__5¬ûÿ 9÷„æûÿÿÿ,¼ç³9
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Ç¢9Gamma1LË£9ÿÿÿÿÿÿÿÿ`æ¤9*creates the congruence subgroup Gamma^0(N)ÈÐðÐ¥9¬ûÿûÿûÿûÿœæÌåæDÔ¦9
__sig__5Õ¸ÕÐÕ§9Ìå˜æûÿÿÿ(Ø¨9
GammaUpper00ÛXÛ©9ÿÿÿÿÿÿÿÿ”æª9*creates the congruence subgroup Gamma^1(N)pãˆã9@The q-expansion of the newform attached to the elliptic curve E.m for the s9&#ûÿûÿûÿûÿ<äÌã8äithe9
__sig__5ic field.9Ìã@äûÿÿÿ9__sig__8genvaluesector9\Exactly the same as CompactSystemOfEigenvalues, but returns only the vector and not the map.m attac9¶ûÿûÿûÿûÿLäÌãHä9
__sig__6ãDä9ÌãPäûÿÿÿX9
__sig__0genvalues9NThe system of Hecke eigenvalues [a2, a3, a5, a7, ..., a_p] attached to M, where p is the largest prime less or equal to prec. The a_i lie in a quotient of a polynomial extension of the base field of M. It is assumed that M corresponds to a single Galois-conjugacy class of newforms.more comp9¶ûÿûÿûÿûÿ\äÌãXä9Ìã`äûÿÿÿØä	95/home/was/magma/packages/modsym/code/representation.mp8ß:áVûÿûÿûÿûÿ„ë î€ë$â:	__sig__21ÿÿ<ã:
ComplexValuePä:ÿÿÿÿÿÿÿÿdëå:áVûÿûÿûÿûÿœë î˜ëç: î ëûÿÿÿé:áþÿáûÿûÿûÿûÿ´ë¬ë î¨ë,êë: î°ëûÿÿÿImagí:ÿáþÿûÿûÿûÿûÿØë¼ë î¸ëð:
	__sig__24ûÿûÿûÿûÿï: îÀëûÿÿÿ__siò:
	È¦	ñ:«þÿ«ûÿûÿûÿûÿ0øÌë îÈëáûÿô:
	__sig__25è	Èè	ó: îÐëûÿÿÿ>

>ÿ«þÿûÿûÿûÿûÿ8ø î,øûÿ>
	__sig__26èè	¨é	> î4øûÿÿÿ>
x!õ:
Ì$ö:««ûÿûÿûÿûÿèëàë,êÔëp5÷:
	__sig__20´Bø:,êÜëûÿÿÿ€Dù:
Dú:««ûÿûÿûÿûÿøëðë,êäë´Kû:
	__sig__21Rü:,êìëûÿÿÿp^ý:
„^þ:ÿ«ûÿûÿûÿûÿìì,êôë¸^ÿ:
	__sig__22Ì^;,êüëûÿÿÿä^;
ô^;ÿ«ûÿûÿûÿûÿ<õì,êì8g;
	__sig__23°v;,êìûÿÿÿ´;
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	__sig__28Œ; î ìûÿÿÿä«
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	__sig__29è²; î0ìûÿÿÿ¶;
h·;ááûÿûÿûÿûÿ,õ<ì î8ì\»;
	__sig__30p»; î@ìûÿÿÿ Ã;Adivision of a point in the upper half plane by a positive integer;ááûÿûÿûÿûÿLì îHì,ê; `returns true if and only if a and b are equivalent under the 	action of G on the upperhalf plane;¬áá­ûÿûÿûÿPø\ì îXì Ê;	__sig__32ÿÿ8Ê;+/usr/local/Magma2.8/package/GrpPSL2/cusps.mÈñ	ö	;+/usr/local/Magma2.8/package/GrpPSL2/cusps.mò	ö	; areturns a list of coset representatives of G in PSL2(Z); only defined for G a subgroup of PSL2(Z)ðÜ;¬ûÿûÿûÿûÿítìêpì$Ý;
__sig__5Ý8Ý;êxìûÿÿÿPÝK;
__sig__1ackages/fun ;./home/was/magma/packages/fundom/farey_gamma0.m	Øô	ë=
	gen_edges ô	Øô	#;WidthsÞ$;ÿÿÿÿÿÿÿÿlìF;widthsd%;"kreturns a list of inequivalent elliptic points of the congruence subgroup G acting on the upperhalf plane Hæ';
__sig__2ê@ê‰94¬Given N = [n,m,p], this returns the congruence subgroup consisting of 2 by 2 matrices with integer coefficients [a,b,c,d] with a = d = 1 mod m, b = 0 mod p, and c = 0 mod n\ÇŒ9PSL2ÊäÊ9ÿÿÿÿÿÿÿÿ÷œ9Gamma0oup9<ÑGiven N = [n,m,p], this returns the congruence subgroup consisting of 2 by 2 matrices with integer coefficients [a,b,c,d] with b = 0 mod p, and c = 0 mod n, and char(a) = 1 for char a Dirichlet character mod mŽ9¯¬ûÿûÿûÿûÿxæ@æ÷<æ„à‘9__sig__2ÿÿÿìâ’9CongruenceSubgroupdä|äø8^The vector corresponding to x in the underlying vector space corresponding to the parent of x.t pri9·uûÿûÿûÿûÿŒäÄóläàã. It&9
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__sig__2or internal)9lä¨äûÿÿÿ,9&~The subspace of the linear dual of Representation(AmbientSpace(M)) that is isomorphic to M as a module over the Hecke algebra.+9¶tûÿûÿ´älä°äÈl.9
__sig__3¶-9lä¸äûÿÿÿ09&|The subspace of the linear dual of Representation(AmbientSpace(M)) that is isomorphic to M as module over the Hecke algebra.inear d/9¶tûÿûÿûÿûÿÄäläÀäÜz29
__sig__4¶19läÈäûÿÿÿ59¶ÔûÿûÿûÿÜäläØä†1:	__sig__27ÿÿØ5:
	__sig__28 element o4:÷ÔèûÿÿÿÀ
É=The determinant of A.-DRKÊ=­þÿûÿûÿûÿûÿ0÷÷$÷7:The identity element of G.The 6:¬­ûÿûÿûÿûÿàè÷Üèûÿ9:
	__sig__31	,¼ç8:÷äèûÿÿÿÌå;:
	h	::­­ûÿûÿûÿûÿ$ìðè÷ìè=:
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¬cZ:ÿ«­ûÿûÿûÿûÿŒépé÷léûÿ]:
	__sig__38H		
	\:÷téûÿÿÿ_:
Lm^:­­ûÿûÿûÿûÿ€é÷|éûÿa:
	__sig__39(
	è
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pwb:
Dx\xc:ÿ­ûÿûÿûÿûÿ¤ë”éÌåˆéÄ|d:
	__sig__46~°~e:Ìåéûÿÿÿ€f:­ûÿûÿûÿûÿ|ô é÷œé­ûÿi:
	__sig__40è	¨	h:÷¤éûÿÿÿ‡:
lDomaink:5returns the intersection of two congruence subgroups.j:¬¬¬ûÿûÿûÿûÿ°é÷¬éˆ	m:
	__sig__41šŒ:"o:5returns the intersection of two congruence subgroups.4žn:¬¬¬ûÿûÿûÿûÿôôÀé÷¼é|¢q:
	__sig__42t¤Œ¤p:÷Äéûÿÿÿ,¦s:%Returns a sequence of generators of Gence sr:¬ûÿûÿûÿûÿpíÐé÷Ìéôôu:
	__sig__43ç	__sit:÷Ôéûÿÿÿûÿÿÿw:.returns a sequence of points in the upperhalf plane union cusps, such that the geodesics between these points form the boundary of a fundamental domain for G µy:
	__sig__44Ð¶·x:÷äéûÿÿÿP¹z:.returns a sequence of points in the upperhalf plane union cusps, such that the geodesics between these points form the boundary of a fundamental domain for G<¿{:à¬ûÿûÿûÿûÿØéôéÌåèé´Å|:
	__sig__52XÈ`Ê}:Ìåðéûÿÿÿ¨Ì~:FundamentalDomain¸ÎÐÎ:ÿÿÿÿÿÿÿÿìé:Breturns the fixed points of the action of g on the upperhalf planeuch that €:à­ûÿûÿûÿûÿê÷êƒ:
	__sig__45‚:÷êûÿÿÿˆàC;1d'()*8é=__sig__4eGamma0	0u	ä»ç39DualRepresentationtŠ,‹49ÿÿÿÿÿÿÿÿ¬ä69@ãA basis over the integers for the integral modular symbols in the VectorSpace(M). This is the intersection of M with the Z-lattice generated by all modular symbols X^iY^(k-2-i){a,b}. The base field of M must be RationalField().Ôä„š89
__sig__5›¼›Ô›79läàäûÿÿÿ¬99
DualVectorSpace,Ÿ:9ÿÿÿÿÿÿÿÿ¼äB9
__sig__7atione int<9DõA basis over the integers for the integral modular symbols in the vector space representation of M. This is the intersection of M with the Z-lattice generated by all modular symbols X^iY^(k-2-i){a,b}. The base field of M must be RationalField().;9¶ÔûÿûÿûÿûÿôäläðäF9//home/was/magma/packages/modsym/code/subspace.m@´@9(‚A basis over the integers for the integral modular symbols in the DualVectorSpace(M). The base field of M must be RationalField().ith the Z?9¶Ôûÿûÿûÿåläåld()A9läåûÿÿÿ"9The zero subspace of M.åˆäG9
__sig__0_sig__6Åò<ååûÿÿÿ8ÈC9IntegralRepresentationHÌD9ÿÿÿÿÿÿÿÿìäH9¶¶ûÿûÿûÿûÿÈóåˆäe(M)N9AThe cuspidal subspace of M. This is the kernel of BoundaryMap(M).¶M9¶¶ûÿûÿûÿûÿŒö<åå8å°ÜI9
DualLattice à¸àJ9ÿÿÿÿÿÿÿÿüä9CompactSystemOfEigenvaluesÌÓÔ9ÿÿÿÿÿÿÿÿ¸ó95/home/was/magma/packages/modsym/code/representation.mûÿûÿñ<lätäûÿÿÿàã¡8¶ûÿûÿûÿûÿŒâ0áˆâ rep¯80áÀâûÿÿÿnifi²8(†The action of the Atkin-Lehner involution on the dual representation of M, when it is defined (see the documentation for AtkinLehner.)lHeck±8¶ûÿûÿûÿûÿÌâ0áÈâ¼çÃ9
ntGrpPSL2¥8DualHeckeOperator(P¦8ÿÿÿÿÿÿÿÿlâ®8)The same as DualAtkinLehnerOperator(M,q).ˆ	0 	­8¶ûÿûÿûÿûÿ¼â0á¸âˆ 	°8
__sig__9 0 	ˆ 	º8
	__sig__11ratorl"«8
AtkinLehnerÜ#$¬8ÿÿÿÿÿÿÿÿ„â´8
	__sig__10 that repr³80áÐâûÿÿÿn M,Ä8bkMultiplication by X^pY - XY^p, which is a possible analogue of the theta-operator. (On mod p modular forms, the theta-operator is the map given by f |--> q df/dq.) Both M_1 and M_2 must be spaces of modular symbols over a field of positive characteristic p; they must have the same level and character, and the weight of M_2 must equal the weight of M_1 plus p+1.des q.O8ZGA surjective linear map from the ambient space of M to a vector space, such that the kernel of this map is the same as the kernel of the period mapping. Note that M must be defined over the rational numbers. This map is normalized so that the image of IntegralBasis(CuspidalSubspace(AmbientSpace(M))) is the standard Z-lattice.ûÿÌ8¶ûÿûÿûÿûÿ¤ó$ã<áHRµ8
DualAtkinLehnerPS¶8ÿÿÿÿÿÿÿÿ´âÀ8
	__sig__12rator¨W¸8"hThe conjugation involution * on the dual representation of M (see the documentation for StarInvolution.)cumentation·8¶ûÿûÿûÿûÿäâ0áàâûÿ¹80áèâûÿÿÿ__siÆ8
	__sig__13<á»8DualAtkinLehnerOperatorg¼8ÿÿÿÿÿÿÿÿÄâ¾8 dThe conjugation involution * on M that sends the modular symbol X^iY^j{u,v} to (-1)^jX^iY^j {-u,-v}.n.)p½8¶ûÿûÿûÿûÿüâ0áøâ<á¿80áãûÿÿÿ¨wÁ8DualStarInvolution(yLyÂ8ÿÿÿÿÿÿÿÿÜâÃ8¶¶ûÿûÿûÿûÿã0áã-1)^Å80áãûÿÿÿÉ8-/home/was/magma/packages/modsym/code/period.m	Ð)	Ë8
__sig__0_sig__12é<$ã,ãûÿÿÿÔÇ8
StarInvolutionøÈ8ÿÿÿÿÿÿÿÿôâö8¶#ûÿûÿûÿûÿ4ä¨óÌã@ãr. (û8Ìãðãûÿÿÿs th9DöThe K-vector space basis of q-expansions in reduced row echelon form for the sum of the spaces of modular forms associated to the modular symbols spaces in D, where K is their common base field. The absolute precision of the q-expansions is prec.ÿ8ÿ¶ûÿûÿûÿûÿääÌãä39
__sig__2á,ãûÿÿÿÍ8
ThetaOperator¸­Î8ÿÿÿÿÿÿÿÿãÏ8+/usr/local/Magma2.8/package/ModSym/period.mè,	1	ê<ÌãÔãûÿÿÿcodeÒ8.˜A surjective linear map from the ambient space of M to a vector space, such that the kernel of this map is the same as the kernel of the period mapping.Note that MÑ8¶ûÿûÿûÿûÿLã$ãHãs(CuÔ8
__sig__1))) is the Ó8$ãPãûÿÿÿÙ8$ãhãûÿÿÿÝ8¶uûÿûÿûÿûÿŒã|ã$ãxã@ãÕ8
IntegralMapping(ÐÖ8ÿÿÿÿÿÿÿÿ0ãØ80ŸThe dual modular symbol associated to M, viewed as a map of Hecke modules M_k ----> M_k/Ker(Phi_M), where the quotient is viewed as an abstract *vector space.*XÛ0	è0	÷<hecke0Û8
RationalMapping â<IROì4<ÿÿÿÿÿÿÿÿDð<*Left ideal of A with generator sequence S.Œ¤<§§ûÿûÿûÿûÿdð|îXð<
	__sig__24ü<|î`ðûÿÿÿ`<
	LeftIdealD\<ÿÿÿÿÿÿÿÿ\ð<+Right ideal of A with generator sequence S.Ôì<§§ûÿûÿûÿûÿ|ð|îpð<
	__sig__25<|îxðûÿÿÿ€ <

RightIdeal<!<ÿÿÿÿÿÿÿÿtð"<"The unique two-sided ideal over p.¨#À##<§§ûÿûÿûÿûÿ”ð|îˆðD($<
	__sig__26œ*¼*%<|îðûÿÿÿì,&<

PrimeIdeal/l/'<ÿÿÿÿÿÿÿÿŒð0<Unitsl,6(<;The ideal of A generated by elements of the form x*y - y*x.ô9:)<§§ûÿûÿûÿûÿ¬ð|î ðì<*<
	__sig__27D>l>+<|î¨ðûÿÿÿÔ?,<6The units in the quaternion order S over the integers.D-<§ûÿûÿûÿûÿ¼ð|î°ð¤H.<
	__sig__28üIJ/<|î¸ðûÿÿÿPM1<ÿÿÿÿÿÿÿÿ´ð2<
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	__sig__29ÜUV5<|îÐðûÿÿÿX6<7/usr/local/Magma2.8/package/AlgQuat/quaternion_ideals.mØE	7<7/usr/local/Magma2.8/package/AlgQuat/quaternion_ideals.mØE	S<
__sig__1esa(a8<KA sequence of representatives for the 2-sided ideal classes of the order A.Èdàd9<§ûÿûÿûÿûÿìðÜðàðÈg:<
__sig__0k`kˆk;<ÜðèðûÿÿÿÈl<<TwoSidedIdealClassesnn=<ÿÿÿÿÿÿÿÿäð><2Representatives for the right ideals classes of A.¼uÔu?<§ûÿûÿûÿûÿñÜðøðHy@<
__sig__1zlz”zA<Üðñûÿÿÿd}B<RightIdealClasses4C<ÿÿÿÿÿÿÿÿüðD<1Representatives for the left ideals classes of A.8ˆ ˆE<§ûÿûÿûÿûÿñÜðñÐF<
__sig__2ÜôG<Üðñûÿÿÿ˜’H<LeftIdealClasses“à“ø“I<ÿÿÿÿÿÿÿÿñJ<2/usr/local/Magma2.8/package/AlgQuat/isomorphisms.m€>	ØE	K<2/usr/local/Magma2.8/package/AlgQuat/isomorphisms.mÐ>	ØE	L<QuaternionIsomorphismð ý$k9T+/home/was/magma/packages/modss/code/modss.mà?	`@	õ<
	ss_pointsO<
__sig__0­l­„­P<,ñ<ñûÿÿÿ¯Q<NTrue if and only if I is isomorphic to J as an ideal over a common left order.”²R<§§ûÿûÿûÿûÿPñ,ñDñœµT<,ñLñûÿÿÿ”·U<IsLeftIsomorphic¹d¹|¹V<ÿÿÿÿÿÿÿÿHñW<OTrue if and only if I is isomorphic to J as an ideal over a common right order.¿ ¿8¿X<§§ûÿûÿûÿûÿhñ,ñ\ñ°ÅY<
__sig__2ÈTÈ\ÊZ<,ñdñûÿÿÿ¤Ì[<IsRightIsomorphic´ÎÌÎ\<ÿÿÿÿÿÿÿÿ`ñ^<§§ûÿûÿûÿûÿ€ñ,ñtñ Ô_<
__sig__3ÕôÕtÖ`<,ñ|ñûÿÿÿTØa<
`Ü¬Üb<§§šûÿûÿûÿñ,ñ„ñ„ãc<
__sig__4äå$åd<,ñŒñûÿÿÿ`çe<
LeftIsomorphism êg<

<ëtëŒëŒì&;à¬ûÿûÿûÿûÿ°ì ìhì”ìd(;hìœìûÿÿÿ+;"ireturns a list of inequivalent elliptic points of the congruence subgroup G acting on the upperhalf planeè-;
__sig__3Øð.;hì´ìûÿÿÿ/;8Âfor a congruence subgroup G, finds whether the cusps a and b are equivalent under the action of G, and if so returns true and a matrix in G taking a to b. If not, returns false and the identity.d|0;««¬­ûÿûÿûÿÐìÈìhì¼ì¼1;
__sig__4@è2;hìÄìûÿÿÿè!3;8Âfor a congruence subgroup G, finds whether the cusps a and b are equivalent under the action of G, and if so returns true and a matrix in G taking a to b. If not, returns false and the identity.,,,4;¬­ûÿûÿûÿTìØìhìÌìt15;
__sig__54”5ì56;hìÔìûÿÿÿ¼77;OFor a congruence subgroup G and a cusp x, returns the width of x as a cusp of GÌ;ä;ü;8;«¬ûÿûÿûÿûÿøìèìhìÜìD?9;
__sig__6@@4@:;hìäìûÿÿÿC;;
	CuspWidth,EDE<;ÿÿÿÿÿÿÿÿàì=;OFor a congruence subgroup G and a cusp x, returns the width of x as a cusp of GPMhMˆM>;¬ûÿûÿûÿûÿíhìôì`Q?;
__sig__7RŒR¤R@;hìüìûÿÿÿ<TA;2/usr/local/Magma2.8/package/GrpPSL2/farey_symbol.mXM	@Z	{;*/home/was/magma/packages/fundom/graphics.my¨M	øM	Èô	ä»çD;cosetsorsG;
groupd'(E;

otherEdges `¸`î=__sig__0shN	˜N	Èô	ä»çI;?Computes the Farey Symbol of a congruence subgroup in PSL_2(Z).@eXepeH;¬æûÿûÿûÿûÿ(í„ì$í˜jJ;„ì,íûÿÿÿ¸O	U;
__sig__3yxmL;
FareySymbolänŒoM;ÿÿÿÿÿÿÿÿ´÷O;
mputes theN;æûÿûÿûÿûÿûÿ@í„ì<íQ;
__sig__2ûÿûÿûÿûÿP;„ìDíûÿÿÿ__si[;
__sig__4íö;
teeraS;+Returns the list of cusps of a Farey SymbolR;æûÿûÿûÿûÿPí„ìLíT;„ìTíûÿÿÿXV;HackobjPrintSymFryÜôW;ÿÿÿÿÿÿÿÿ8íY;3Returns the list of edge labels of a Farey Sequence°R	`S	X;æûÿûÿûÿûÿhí„ìdí¸S	Z;„ìlíûÿÿÿ];SReturns the generators of the congruence subgroup corresponding to the Farey SymbolP¢x¢\;æûÿûÿûÿûÿxí„ìtí ¦_;
__sig__5¨(¨x¨^;„ì|íûÿÿÿ€ªa;DReturns the congruence subgroup corresponding to the Farey Symbol FSe Farey`;æ¬ûÿûÿûÿûÿˆí„ì„íc;
__sig__6¼ç__sib;„ìŒíûÿÿÿûÿÿÿe;7Returns the widths of the cusp list of the Farey Symbolnding to thed;æûÿûÿûÿûÿ˜í„ì”íûÿg;
__sig__7W	,¼çf;„ìœíûÿÿÿíy;
	__sig__11es¨	i;]Returns the coset representatives for the congruence subgroup of the Farey Symbol in PSL_2(z)h;æûÿûÿûÿûÿ¨í„ì¤í0X	k;
__sig__8¤Ïs;
	__sig__10esÑ<Ñm;4³Returns a sequence of pairs of cusps which are cusps of the Farey Symbol FS, and which are not adjacent in FS but which are images of 0 and infinity under some matrix in PSL_2(Z).l;æûÿûÿûÿûÿ¸í„ì´íLâp;æûÿûÿûÿûÿÈí„ìÄípær;„ìÌíûÿÿÿ0èÎ;
__sig__9k	½	øÏ;|î8ïûÿÿÿÐ;

NormModule$Ñ;ÿÿÿÿÿÿÿÿ4ïÜ;
	__sig__12nomialÒ;#The characteristic polynomial of x.xÓ;šûÿûÿûÿûÿ\ïTï|îHï¬Ô;
	__sig__10ÀÕ;|îPïûÿÿÿØÖ;#The characteristic polynomial of x.ø×;Üûÿûÿûÿûÿdï|îXïÄØ;
	__sig__11ØÙ;|î`ïûÿÿÿðÚ;=The coordinates of x with respect to the basis of its parent.¸	Û;šûÿûÿûÿûÿtï|îhïì	Ý;|îpïûÿÿÿ
Þ;"lThe conductor of the quaternion order, defined to be the index in a maximal order of its quaternion algebra.H
ß;§ûÿûÿûÿûÿ„ï|îxïd!à;
	__sig__13€!á;|î€ïûÿÿÿØ$â;True if K is isomorphic to L.d5ã;WWûÿûÿûÿûÿ8ñ”ï|îˆïtDä;
	__sig__14Då;|îïûÿÿÿ˜Kæ;"hTrue if and only if p is a ramified prime in K, where K must be a quaternion algebra over the rationals.RäYç;Wûÿûÿûÿûÿ¤ï|î˜ï¤^è;
	__sig__15¸^é;|î ïûÿÿÿÐ^ê;"gTrue if and only if K is ramified at infinity, where K must be a quaternion algebra over the rationals. g$gë;Wûÿûÿûÿûÿ´ï|î¨ï¼vì;
	__sig__16œƒí;|î°ïûÿÿÿÈ‡î;

IsDefiniteä‡ï;ÿÿÿÿÿÿÿÿ¬ïð; dTrue if and only if K is split at infinity, where K must be a quaternion algebra over the rationals.¬ñ;WûÿûÿûÿûÿÌï|îÀïè²ò;
	__sig__17üµó;|îÈïûÿÿÿl·ô;
IsIndefinite€·õ;ÿÿÿÿÿÿÿÿÄï÷;§ûÿûÿûÿûÿäï|îØï¬Ãø;
	__sig__18ÀÃù;|îàïûÿÿÿØÃú;:A Minkowski-reduced basis for A, unique up to isomorphism.˜Éû;§ûÿûÿûÿûÿôï|îèïÌÉü;
	__sig__19àÉý;|îðïûÿÿÿøÉþ;JA canonical Minkowski-reduced Gram matrix for the norm inner product on A.0Êÿ;§ûÿûÿûÿûÿð|îøï×<
	__sig__200×<|îðûÿÿÿ¼Ü<ReducedGramMatrixØÜ<ÿÿÿÿÿÿÿÿüï<§§§ûÿûÿûÿûÿð|îð,Ý<
	__sig__21@Ý<|îðûÿÿÿXÝ<IMThÝ	<ÿÿÿÿÿÿÿÿð
<
˜Ý<§§ûÿûÿûÿûÿ4ð|î(ðÌÝ<
	__sig__22àÝ<|î0ðûÿÿÿøÝ<ILOÞ<ÿÿÿÿÿÿÿÿ,ð<
Ðå<§§ûÿûÿûÿûÿLð|î@ðæ<
	__sig__234ê<|îHðûÿÿÿí:
#$%&“š:

laneWithCusps@Å“:VàáûÿûÿûÿûÿØ÷Tê îPênary•: îXêûÿÿÿðl	ó=
HalfPlaneEltô=ÿÿÿÿÿÿÿÿLêõ=
hÓ€Ó÷=__sig__2ÿÿÿ8Õ—:UpperHalfPlaneWithCuspsD×˜:ÿÿÿÿÿÿÿÿ(î™:àûÿûÿûÿûÿûÿlê îhêìâœ:
__sig__7ãôãä›: îpêûÿÿÿ æ¨:
__sig__9pElt0è:CosetRepresentativesÚÛ:ÿÿÿÿÿÿÿÿèç:6returns true if and only f G is a congruence 	subgroup:÷èûÿÿÿûÿ:	__sig__24	(Á	ÔH	,î9

IsSubgroupï9ÿÿÿÿÿÿÿÿ˜çó9
	__sig__19her G is fò9÷Ìçûÿÿÿ÷90returns index of G in SL_2(Z), if this is finiteåÀçù9
	__sig__20¤ø9÷äçûÿÿÿ¼ô9
IsFiniteIndexÐõ9ÿÿÿÿÿÿÿÿ¨çö9¬ûÿûÿûÿûÿØíàç÷Üç:
	__sig__22esûÿû9 areturns a list of coset representatives of G in PSL2(Z); only defined for G a subgroup of PSL2(Z)øq	èr	ú9¬ûÿûÿûÿûÿ íðç÷ìçý9
	__sig__21ûÿü9÷ôçûÿÿÿ,¼çÿ9The level of G.Ìå:
	__sig__23	0£	þ9¬ûÿûÿûÿûÿ´õè÷üçbgro:÷èûÿÿÿ:¬ûÿûÿûÿûÿè÷è__si:@returns true if and only if G is equal to Gamma_0(N) 	for some Nåèûÿÿÿ:(ƒFor G a congruence subgroup, returns [[N,M,P]] where G consists of matrices [a,b,c,d] with c = 0 mod N, a, d = 1 mod M, b = 0 mod PÀK	:¬ûÿûÿûÿûÿ,èÌå èˆ^
:
	__sig__25œ^:Ìå(èûÿÿÿ´^:CongruenceIndicesÌ^:ÿÿÿÿÿÿÿÿ$è:¬ûÿûÿûÿûÿ@è÷<è:÷Dèûÿÿÿûÿ:@returns true if and only if G is equal to Gamma_1(N) 	for some NÌå:
IsCongruenceä‡:ÿÿÿÿÿÿÿÿè:¬ûÿûÿûÿûÿXè÷TèPy	:
	__sig__25:÷\èûÿÿÿÌå:IThe genus of the upper half plane quotiented by the congruence subgroup GHy	z	:¬ûÿûÿûÿûÿhè÷dè:
	__sig__26:÷lèûÿÿÿ€»/:7returns an element of a matrix group corresponding to gÄÃ:@returns true if and only if G is equal to Gamma^0(N) 	for some NÉˆÉ:¬ûÿûÿûÿûÿ|èÌåpè¼É:
	__sig__29ÐÉ:ÌåxèûÿÿÿèÉ!:ÿÿÿÿÿÿÿÿtè?:
er1Ê":@returns true if and only if G is equal to Gamma^0(N) 	for some NÔPÕ#:¬ûÿûÿûÿûÿ”èÌåˆèŒÜ$:
	__sig__30ÄÜ%:ÌåèûÿÿÿÜÜ&:
IsGammaUpper0ðÜ':ÿÿÿÿÿÿÿÿŒè(:@returns true if and only if G is equal to Gamma^1(N) 	for some NÝ@Ý):¬ûÿûÿûÿûÿ¬èÌå ètÝ*:
	__sig__31ˆÝ+:Ìå¨èûÿÿÿ Ý,:
IsGammaUpper1´Ý-:ÿÿÿÿÿÿÿÿ¤è.:­kûÿûÿûÿûÿÀñÀè÷¼èÞ0:÷Äèûÿÿÿ2:¬ûÿûÿûÿûÿÐè÷Ìè»z9
gammaType_listy9

MemberTest(½@½|9subgroup_list{9
conjugate_listŒÀ~9
cusp_widths}9cusps`Ç€9
	FS_labels¡;+9elliptic_points0ÎHÎ Îƒ9
	FS_cosets„9
FS_gen_edges9
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__sig__1ctive speciŠ9÷,æûÿÿÿÈ‚	(	$	…9
	FS_widths0ÛXÛ†9
FS_otherEdgesHßu9\" !6‡9n7„:'/home/was/magma/packages/fundom/cusps.mÀ„	†:ûÿûÿûÿûÿH÷êêûÿÑ=
__sig__0„	À„	Ò=êD÷ûÿÿÿÕ=
ðˆ:
FixedPoints\t‰:ÿÿÿÿÿÿÿÿêŠ:6/usr/local/Magma2.8/package/GrpPSL2/upper_half_plane.m	ð—	4 fThe congruence group of space of cusp forms corresponding to the space of cuspidal modular symbols M. Let S=S_k(Gamma_0(N),Z), let V be the sub Z-module corresponding to M, and W be its orthogonal complement. Then the congruence group is S/(V+W). This group encodes information about congruences between forms in V and forms in the complement of V. The optional parameter Bound is a bound on the number of Hecke operators used in the computation; if it is too small, then Congruence group will give only an upper bound on the correct answer. The default is HeckeBound(M) + 1, which gives a provably correct answer.ä6$vThe intersection pairing (x,y) of the homology classes corresponding to the weight-2 cuspidal modular symbols x and y.Øã<ôÛüÛûÿÿÿ,!7l“The degneracy map M_1 ---> M_2 associated to d. Let N_i be the level of M_i for i=1,2. Suppose that d is a divisor of either the numerator or denominator of the rational number N_1/N_2, written in reduced form. If N_1 divides N_2, then this intrinsic returns alpha_d : M_1 ---> M_2, or if N_2 divides N_1, then this intrinsic returns beta_d : M_1 ---> M_2. It is an error if neither divisibility holds. ¤7¼77dvThe modular symbols space of level N associated to M. Let NN be the level of M. If NN divides N, then this intrinsic returns the modular symbols space Sum_(d|N/NN)alpha_d(M) If N divides NN, then this intrinsic returns the modular symbols space Sum_(d|NN/N)beta_d(M). In this latter case, if Conductor(DirichletCharacter(M)) does not divide N, then the 0 space is returned. $G8t±A positive integer n such that the Hecke operators T1,...,Tn generate the Hecke algebra as a Z-module. When the character is trivial, the default bound is (k/12)*[SL_2(Z):Gamma_0(N)]. When the character of M is nontrivial, the default bound is twice the above bound; however, it is not known that this bound is large enough in all cases in which the character is nontrivial, so one may wish to increase the bound using SetHeckeBound.<[T[9ÌThe K-vector space basis of q-expansions in reduced row echelon form for the space of modular forms associated to M, where K is the base field of M. The absolute precision of the q-expansions is prec. The optional parameter Al can take the values "Newform" and "Universal". The default is "Newform", which computes a basis of q-expansions by finding a decomposition of M into subspace corresponding to newforms, computing their q-expansions, and then taking all of their images under the degeneracy maps. If Al is set equal to "Universal" then the algorithm described in Merel's paper "Universal Fourier expansions of modular forms" is used instead. This latter algorithm does not require computing a newform decomposition of M, but requires computing the action of more Hecke operators.0}d}|xøü $(,048<@DHLPTX\`dhlptx|€„ˆŒ”˜œ ¤¨¬°´¸¼ÀÄÈÌÐÔØÜàäèìðôø˜œ ¤¨¬°´¸¼ÀÄÈÌÐÔØÜàäèìðôøüœ	 	¤	¨	¬	°	´	¸	¼	À	Ä	È	Ì	Ð	Ô	Ø	Ü	à	ä	è	ì	ð	ô	ø	ü	







 
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ÔÜàäèà è d!l!p!t!x!€!„!È$Ì$Ð$Ô$Ø$Ü$à$ä$H5\5`5d5h5l5p5t5x5|5€5´BXDdDlDpDtD€D„DˆDŒDD”D„KŒKK”K˜KœK K¤K¨K¬K°K´K¸K¼KÀKRRRR R(RäYp^x^|^€^„^ˆ^Œ^^”^˜^œ^ ^¤^¨^¬^°^´^¸^¼^À^Ä^È^Ì^Ð^Ô^Ø^Ü^à^ä^è^ì^ð^ô^ø^ü^____ g$g(g,g0g4g8g<gPnXn¨v°v´v¸v¼vÐy¬´¸œƒ¤ƒ¨ƒ¬ƒ°ƒ´ƒÈ‡Ð‡Ô‡Ü‡à‡ä‡@ŠHŠLŠPŠTŠXŠ\ŠˆŒÐ«Ø«Ü«à«ä«è«ì«ð«ô«ø«ü«¬¬¬¬¬¼²Ä²È²Ì²Ð²Ô²Ø²Ü²à²ä²è²ì²ð²ô²ø²üµ¶¶`·d·h·l·p·t·x·|·€·,¹4¹8¹L»T»X»\»`»d»h»l»p»t»x»|»€»˜Ã Ã¤Ã¨Ã¬Ã°Ã´Ã¸Ã¼ÃÀÃÄÃÈÃÌÃÐÃÔÃØÃÜÃàÃäÃìÃ|É„ÉˆÉŒÉÉ”É˜ÉœÉ É¤É¨É¬É°É´É¸É¼ÉÀÉÄÉÈÉÌÉÐÉÔÉØÉÜÉàÉäÉèÉìÉðÉôÉøÉüÉÊÊÊÊÊ	ÊÊ Ê$Ê(Ê,Ê0Ê4Ê8Ê<Ê@ÊDÊÔPÕXÕ×××××× ×$×(×0×ˆÜŒÜÜ”Ü˜Ü¼ÜÄÜÈÜÌÜÐÜÔÜØÜÜÜàÜäÜèÜìÜðÜôÜøÜüÜÝÝÝÝÝÝÝÝ Ý$Ý(Ý,Ý0Ý4Ý8Ý<Ý@ÝDÝHÝLÝPÝTÝXÝ\Ý`ÝdÝhÝlÝpÝtÝxÝ|Ý€Ý„ÝˆÝŒÝÝ”Ý˜ÝœÝ Ý¤Ý¨Ý¬Ý°Ý´Ý¸Ý¼ÝÀÝÄÝÈÝÌÝÐÝÔÝØÝÜÝàÝäÝèÝìÝðÝôÝøÝüÝÞÞÞÞÞÞÞÞ Þ$Þ(Þ,Þ4Þ8ÞÐåØåÜåàåäåèåìåðåôåøåüåæææææææ4ê8ê@êDêHêíííí€îˆîî”îÐóÔóØóÜóàóäóèó÷÷¬÷¨ƒ¬ƒ=
__sig__1‡Ô‡=¬åôûÿÿÿLŠ=´ûÿûÿûÿûÿ$ô¬åôè«=
__sig__2«ü«	=¬å ôûÿÿÿ¼²
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UsesBrandtÔ²=ÿÿÿÿÿÿÿÿô=
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__sig__3»d»=¬å8ôûÿÿÿ|»=

UsesMestre¨Ã=ÿÿÿÿÿÿÿÿ4ô=
ØÃ=´8ûÿûÿûÿTô¬åHô É=
__sig__4É´É=¬åPôûÿÿÿÌÉ=
ceeÜÉ=
ìÉ=´ûÿûÿûÿûÿdô¬åXô$Ê=
__sig__5Ê8Ê=¬å`ôûÿÿÿXÕ=
×=´ûÿûÿûÿûÿtô¬åhôÄÜ=
__sig__6ÜØÜ=¬åpôûÿÿÿðÜ=µûÿûÿûÿûÿ„ô¬åxô$Ý =
__sig__7Ý8Ý!=¬å€ôûÿÿÿPÝ"=
`Ý#=´ûÿûÿûÿûÿ”ô¬åˆô”Ý$=
__sig__8Ý¨Ý%=¬åôûÿÿÿÀÝ,=
larPoints&=
àÝ'=´ûÿûÿûÿûÿ¤ô¬å˜ôÞ(=
__sig__9Þ(Þ)=¬å ôûÿÿÿÜå*=SupersingularPointsôå+=ÿÿÿÿÿÿÿÿœô-=´µûÿûÿûÿûÿ¼ô¬å°ôî©;|î îûÿÿÿÅª; A maximal quaternion order in K.ÈTÈ\Ê«;W§ûÿûÿûÿûÿ´î|î¨îüÎ¬;
__sig__1Ð€Ð˜Ð­;|î°îûÿÿÿPÒ®; _Quaternion order generated by S, preserving S if it constitutes a basis with initial element 1.¬ÖÄÖÜÖ¯;ÿš§ûÿûÿûÿûÿÌîÄî|î¸îDß°;
__sig__2àŒáâ±;|îÀîûÿÿÿTã²;+Returns a quaternion order of level M in K.ˆæ æ³;W§ûÿûÿûÿûÿÜîÔî|îÈîŒì¤ììì);
EllipticPoints*;ÿÿÿÿÿÿÿÿ˜ì,;¬ûÿûÿûÿûÿ¸ìhì¬ì‹:6/usr/local/Magma2.8/package/GrpPSL2/upper_half_plane.m	HÂ	:
ScalarFieldPŽ:
isExactt[;:isCusp;¦:
ntSpcHyp‘:
complex_value’:
exact_valueÜ\;:HackobjCoerceSpcHyp ž:ÿÿÿÿÿÿÿÿ øŸ:áûÿûÿûÿûÿûÿ„ê î€ê ¢:
__sig__8^;¡: îˆêûÿÿÿ§^;£:HackobjPrintSpcHyp¤:ÿÿÿÿÿÿÿÿdê®:
	__sig__10Eltd¥:àáûÿûÿûÿûÿ´ôœê î˜ê§: î êûÿÿÿ°³	²:
	__sig__11©:HackobjPrintSpcHypElt¯^;ª:ÿÿÿÿÿÿÿÿ|ê¬: Assign names to elliptic points.«:ÿàûÿûÿûÿûÿûÿ´ê î°ê,ê°:An elliptic point. Assi¯:à­ûÿûÿûÿûÿÄê îÀê±: îÈêûÿÿÿàµ	¸:
pcHyp´:
Äêûÿÿÿ³:àûÿûÿûÿûÿÔê îÐê¶:
	__sig__12ûÿûÿûÿûÿµ: îØêûÿÿÿ__si¾:<checks whether two hyperbolic spaces have the same dimension
pcHypD-;·:áàûÿûÿûÿûÿäê îàêûÿº:
	__sig__13·	Ð·	¹: îèêûÿÿÿÄ:
ïY;»:
HackobjInSpcHyp0¦¼:ÿÿÿÿÿÿÿÿÌêÆ:
	__sig__15pElt;½:ààûÿûÿûÿûÿëüê îøêd’À:
	__sig__14Z;¿: îëûÿÿÿÉ‰f;Á:HackobjParentSpcHypEltÂ:ÿÿÿÿÿÿÿÿÜêÃ:ááûÿûÿûÿûÿÔôë îëÅ: îëûÿÿÿÈ::returns the field over which the matrices of H are definedÇ:àõÿûÿûÿûÿûÿ$ë î ëûÿÊ:
	__sig__16 º	`»	É: î(ëûÿÿÿÌ:KReturns true if and only if the element x of the upper half plane is a cusp H are dË:áûÿûÿûÿûÿ4ë î0ëÎ:
	__sig__17ç	__siÍ: î8ëûÿÿÿûÿÿÿÏ:
ScalarFieldˆÃ:[;Ð:ÿÿÿÿÿÿÿÿëÒ:KReturns true if and only if the element x of the upper half plane is a cusplÈÝÒP;Ñ:áûÿûÿûÿûÿLë îHëÔ:
	__sig__18ÝÒP;Ó: îPëûÿÿÿlÈÖ:J	For x an element of the upperhalf plane, if x is a cusp, returns the value of x as an object of type SetCspElt; if x has an exact value in a quadratic extension, returns this value, as an object of type FldQuadElt; otherwise returns a complex value of type FldPrElt Ðê:
ue an eleÚ: aFor an element of the upper half place, returns the value of x as an element of the complex fieldvalue in aÙ:áVûÿûÿûÿûÿlë îhëex vÜ:
	__sig__20P;0¿	ÀÀ	Û: îpëûÿÿÿà:8Imaginary part of argument, as an element of RealField()´ÑP;á: îˆëûÿÿÿ¤ÐÝ:

ExactValue;Þ:ÿÿÿÿÿÿÿÿTëî:
ueæ:8Imaginary part of argument, as an element of RealField() of x as anè:
	__sig__22K;ì:
	__sig__23w;\returns the index in PSL2(Z) of the congruence subgroup corresponding to the Farey Symbol FSt adjacx;„ìäíûÿÿÿnity–:
__sig__1hCuspst;
InternalEdgesôu;ÿÿÿÿÿÿÿÿ°ív;æûÿûÿûÿûÿàí„ìÜíeodez;*/home/was/magma/packages/fundom/graphics.mmèÃ	`Æ	B;
g to the F};ûÿûÿûÿûÿÀ÷èííûÿ|;
__sig__0Å	,¼çð=èíðíûÿÿÿí‚;æûÿûÿûÿûÿ<îîèíîcorr‰;2/home/was/magma/packages/fundom/upper_half_plane.mxÅ	`Æ	ˆ;D2/home/was/magma/packages/fundom/upper_half_plane.m	ÈÅ	`Æ	ñ=
__sig__0Æ	©	”:
ð*~;2/usr/local/Magma2.8/package/GrpPSL2/farey_gamma0.m`Æ	×	f<ÿÿÿÿÿÿÿÿˆñh<§§šûÿûÿûÿ¨ñ,ñœñi<
__sig__5xj<,ñ¤ñûÿÿÿDk<RightIsomorphismxl<ÿÿÿÿÿÿÿÿ ñm<(/usr/local/Magma2.8/package/Mat/create.mPÝ	ð	n<(/usr/local/Magma2.8/package/Mat/create.m˜Ý	ð	o<*‡The matrix with m rows and n columns, whose rows are given by the elements of Q, where Q is a sequence of m sequences, each of length nèTp<ÿïÿûÿûÿûÿûÿÐñÈñ¸ñ¼ñè!r<¸ñÄñûÿÿÿÀ#s<*ŽThe matrix over R with m rows and n columns, whose rows are given by the elements of Q, where Q is a sequence of m sequences, each of length nü+t<ÿýÿïÿûÿûÿûÿûÿàñØñ¸ñÌñD1u<
__sig__134Ì4v<¸ñÔñûÿÿÿŒ7w<*‡The matrix with m rows and n columns, whose rows are given by the elements of Q, where Q is a sequence of m sequences, each of length nì<==x<ÿïÿûÿûÿûÿûÿðñèñ¸ñÜñ,Ay<
__sig__2BøBCz<¸ñäñûÿÿÿ„E{<*ŽThe matrix over R with m rows and n columns, whose rows are given by the elements of Q, where Q is a sequence of m sequences, each of length nˆM|<ÿýÿïÿûÿûÿûÿûÿøñ¸ñìñ`Q}<
__sig__3RŒR¤R~<¸ñôñûÿÿÿ<T<"nThe diagonal matrix over R with n rows and n columns whose diagonal entries correspond to the the entries of QœZ€<ÿþÿýÿïÿûÿûÿûÿûÿòò¸ñüñh_<
__sig__4`Ð`è`‚<¸ñòûÿÿÿbƒ<$pThe diagonal matrix over R with #Q rows and #Q columns whose diagonal entries correspond to the the entries of Qf¸fÐf„<ÿþÿýÿïÿûÿûÿûÿûÿ òò¸ñò˜l…<
__sig__5mm¨m†<¸ñòûÿÿÿ$p‡<"iThe diagonal matrix with #Q rows and #Q columns whose diagonal entries correspond to the the entries of Q\v€vˆ<ÿþÿïÿûÿûÿûÿûÿ(ò¸ñòz‰<
__sig__6{0{P{Š<¸ñ$òûÿÿÿt~‹<&/usr/local/Magma2.8/package/Tup/misc.m	ð	Œ<&/usr/local/Magma2.8/package/Tup/misc.m	ð	<"Returns the flattend version of T.pŠ(‹Ž<ûÿûÿûÿûÿ@ò0ò4òŒ<
__sig__0‘´‘Ì‘<0ò<òûÿÿÿ(“Å2"The modular forms associated to E.stein.m’<,/home/was/magma/packages/misc/code/Misc.spec©ª•<
magma/packŸ<1/home/was/magma/packages/modform/code/ModFrm.spec/zeta.mÔ<
__sig__0ackages/mod“<)/home/was/magma/packages/misc/code/zeta.m ë	ð	”<)/home/was/magma/packages/misc/code/zeta.mèë	ð	–<¯Ûÿûÿûÿûÿûÿ`òPòTòÔº—<
__sig__0»¼¼˜<Pò\òûÿÿÿ<½™<
4¾|¾š<ÛÿûÿûÿûÿûÿpòPòdòÅ›<
__sig__1Ç$Ç\Çœ<Pòlòûÿÿÿ”Ë<ZetaÌÍž<ÿÿÿÿÿÿÿÿhòÈ2
__sig__0ackages/mod¡<2/home/was/magma/packages/modform/code/arithmetic.mties.m <9/home/was/magma/packages/modform/code/abelian_varieties.mï	ð	¢<2/home/was/magma/packages/modform/code/arithmetic.mt`ï	ð	¤<¹¹¹ûÿûÿûÿûÿ¼È˜òˆòŒò`ç¥<
__sig__0èøé ê¦<ˆò”òûÿÿÿìì(2
ï@ïÂ;
	NormSpace¤ÐÃ;ÿÿÿÿÿÿÿÿüîÄ;PThe inner product subspace with respect to the norm generated by the sequence B.;2/usr/local/Magma2.8/package/GrpPSL2/farey_gamma1.m€ñ	¸	€;./usr/local/Magma2.8/package/GrpPSL2/graphics.m	¸	;./usr/local/Magma2.8/package/GrpPSL2/graphics.m	¸	ƒ;"nDisplay a fundametal domain corresponding to the Farey Symbol FS, with edge identifications and cusps labeled.…;
__sig__1@„;èíîûÿÿÿ†;
DisplayPolygons‡;ÿÿÿÿÿÿÿÿôí ;
	NormSpaceionin‘‹;$rThe set of points in the complex plane which are rational or have positive imaginary part, together with infinity.Ðó	Àô	Š;àûÿûÿûÿûÿÈ÷ î,îõ	ò= îÄ÷ûÿÿÿ :
ntSpcHypQ2
__sig__0`ˆ§<<ÉDÉûÿÿÿœ\2,A finite group that measures all possible congruences (to precision prec) between some integral modular form in M1 and some modular form in M2.$<T[2¸¸“ûÿûÿûÿûÿ¤òPÉpÉ”©<PÉ òûÿÿÿÀª<BçA finite group that measures all possible congruences (to precision prec) between some integral modular form in M1 and some modular form in M2. The optional argument prec must be at least Max(PrecisionBound(M1),PrecisionBound(M2)).ð «<¸¸“ûÿûÿûÿûÿpÒ´òPÉ¨òÜ¬<
__sig__1 Ô !­<PÉ°òûÿÿÿœ"’2
__sig__2$$$<$–2PÉHÊûÿÿÿd&¯<`dA group that measures all possible congruences (to precision prec) between some integral modular form in M1 and some modular form in M2, where the coefficient of q^n of the modular forms are set to 0 if n equals 0 or shares any common factors with the level of M1 or M2. To void nonsense, prec should be at least Max(PrecisionBound(M1),PrecisionBound(M2)).:;°<¸¸“ûÿûÿûÿûÿÈòPÉ¼ò >²<PÉÄòûÿÿÿ¨?³<0/home/was/magma/packages/modform/code/creation.mCðú	
	´<0/home/was/magma/packages/modform/code/creation.mF@û	
	µ<FThe space spanned by all modular forms of all weights over the ring R.ôK¶<ýÿ¸ûÿûÿûÿûÿlÊàòÐòÔò\P¸<ÐòÜòûÿÿÿäQ¯2ÐòÀÊûÿÿÿPS¸2
__sig__7T˜T`U·2ÐòàÊûÿÿÿWÉ2&¹ûÿûÿûÿûÿìòËËÐ[»<Ë ËûÿÿÿÔ]Ð2ËDËûÿÿÿŒ`Ö2¸ûÿûÿûÿûÿûÿôò4Ë\Ë|c½<4ËðòûÿÿÿteÝ24ËxËûÿÿÿ¤fé24Ë¨Ëûÿÿÿ,h¾<ûÿûÿûÿûÿûÿó¸Ë´Ë¬mò2$rThe sequence of characteristic 0 spaces of modular symbols with given sign associated to M, when this makes sense.`uxuÁ<ÌËÔËûÿÿÿÜv3
__sig__0wðw xÂ<ÌÌûÿÿÿèy3ZJUse this intrinsic to find the newforms associated to M with prespecified eigenvalues. Here I is a sequence [< p_1, f_1(x)>,...,< p_n,f_n(x)>] of pairs. Each pair consists of a prime number that does not divide the level of M and a polynomial. This intrinsic returns the set of newforms sum a_n q^n in M such that f_n(a_{p_n})=0. (\w3ÍàÍûÿÿÿøÕ3ÍXÏûÿÿÿœ’J3DñThe smallest integer b such that f + O(q^b) determines any modular form f in M. If the optional paramater Exact is set to false than some integer b such that f+O(q^b) determines any modular form f in M is returned, but b need not be minimal.¤ž¼žÊ<Vÿûÿûÿûÿûÿûÿ0ó”ÐÐä£Ì<”Ð,óûÿÿÿ¥Í<
	BugHunterŒ¦¤¦Î<ÿÿÿÿÿÿÿÿ(óð4
__sig__0ackages/modÐ<//home/was/magma/packages/modsym/code/analytic.m
	Ñ<//home/was/magma/packages/modsym/code/analytic.m
	Ò<*The complex period mapping, computed using n terms of q-expansion. The period map is a homomorphism M --> C^d, where d is the dimension of A.8¶Ó<¶ûÿûÿûÿûÿTóDóHó¨ºÕ<DóPóûÿÿÿ ¼W4¶ûÿûÿûÿûÿdÑDó`Ñ<¿c4¶wûÿûÿûÿûÿ\ó°ÐÑ´Åm4¶ûÿûÿûÿûÿ`ó¨Ñ¤Ñ€ÍØ<¨Ñ°Ñûÿÿÿ`Ïr4¨ÑÌÑûÿÿÿ@Ñ‘4zËThe p-new modular degree of M. Assume that M has trivial character, is p-new, and for notational simplicity the weight of $M$ is $2$. Let r be the (rational) period map. The p-new modular degree of M is the order of the finite quotient r(H_{p-new}) / r(M), where H_{p-new} is the p-new subspace of H = H_1(X_0(N),Z) and N is the level of M. WARNING: This number need not be a square, so we do not take the square root as we would for the usual modular degree.üé$ê’4¨ÑLÒûÿÿÿðìŒ;DisplayFareySymbolDomain\ü\$];ÿÿÿÿÿÿÿÿîŽ;&xDisplay a fundametal domain corresponding to the Farey Symbol of a group G, with edge identifications and cusps labeled.e\ete;¬ûÿûÿûÿûÿDîî8îœj;
__sig__2kll‘;î@îûÿÿÿ|m’;8/usr/local/Magma2.8/package/GrpPSL2/fundamental_domain.mrÐ	˜	“;8/usr/local/Magma2.8/package/GrpPSL2/fundamental_domain.mu(	˜	”;<Õcomputes the intersection in the upper half plane of the two geodesics x, y, where x and y are specified by their end points, given as cusps. If the geodesics intersect along a line the empty sequence is returned.x•;àÿ«ÿ«ûÿûÿûÿûÿlî\îLîPîÄ†–;
__sig__0ˆ¼ˆÔˆ—;LîXîûÿÿÿD˜;GeodesicsIntersectionÔ™;ÿÿÿÿÿÿÿÿTîš;8Åcomputes the intersection in the upper half plane of the two geodesics x, y, where x and y are specified by their end points. If the geodesics intersect along a line the empty sequence is returned.œ›;àÿáÿáûÿûÿûÿûÿtîLîhî\ œ;
__sig__1¡¤¡Ô¡;LîpîûÿÿÿT¤´;
__sig__3¥4¥L¥µ;|îÐîûÿÿÿD§¶;KReturns a maximal order in the rational quaternion algebra of discrinant N.Œ«¤«·;§ûÿûÿûÿûÿìîäî|îØîˆ¯¸;
__sig__4°°°È°¹;|îàîûÿÿÿX²º;YReturns a quaternion order of level M in the rational quaternion algebra of discrinant N.¸µÐµ»;§ûÿûÿûÿûÿôî|îèîhº¼;
__sig__5»¨»À»½;|îðîûÿÿÿà¼¾;6The inner product space of A with respect to the norm.l¿¿;W¾ûÿûÿûÿïï|îøîÇÀ;
__sig__6Ê€ÊèÊÁ;|îïûÿÿÿÍÅ;ÿštûÿûÿûÿûÿ$ïï|îï@ÑÆ;
__sig__7ÒlÒ„ÒÇ;|îïûÿÿÿÔÈ;6The inner product module of A with respect to the normH×É;§8ûÿûÿûÿ,ï|î ï˜ßÊ;
__sig__8â8âPâË;|î(ïûÿÿÿˆãÌ;6The inner product space of A with respect to the norm.4çÍ;§8ûÿûÿûÿûÿ<ï|î0ï\íÔíî»=µµµûÿûÿûÿûÿôö¬åèöh¼=
	__sig__41´Ô½=¬åðöûÿÿÿô¾=-/home/was/magma/packages/modss/code/verbose.m	8A	À=SupersingularModule8P…:'/home/was/magma/packages/fundom/cusps.m .	ï=„ì¸÷ûÿÿÿcusps.mce_sÁ=6/home/was/magma/packages/fundom/congruence_subgroups.m	8A	Â=\6/home/was/magma/packages/fundom/congruence_subgroups.m	8A	Ã=CosetRepresentatives.	8A	Ä=
Determinants0!Å=4The projective special linear matrix group PSL(2,R).$<$Æ=ýÿ¬ûÿûÿûÿûÿ ÷÷÷¨(Ç=
__sig__0*+¸+È=÷÷ûÿÿÿ.ˆ9¬ûÿûÿûÿûÿ8æ(æ÷$æÐ49÷Dæûÿÿÿ7Ë=
	__sig__29h9 9Ì=÷,÷ûÿÿÿ;Í=The base ring of G.H<`<Î=¬ýÿûÿûÿûÿûÿõ@÷÷4÷¨?Ï=
	__sig__30HAhAÐ=÷<÷ûÿÿÿ Dl:÷´éûÿÿÿèEv:¬ûÿûÿûÿûÿÀíàé÷ÜéxJ;./home/was/magma/packages/fundom/farey_gamma1.m\N";./home/was/magma/packages/fundom/farey_symbol.m4Q!;./home/was/magma/packages/fundom/farey_symbol.mPSì=?Computes the Farey Symbol of a congruence subgroup in PSL_2(Z).WXWxWí=¬æûÿûÿûÿûÿ í¼÷„ì°÷\j;„ì¬íûÿÿÿD^o;
__sig__9`t`Œ`n;„ì¼íûÿÿÿ¼aq;:Ëreturns a sequence of points in the upperhalf plane union cusps, such that the geodesics between these points form the boundary of a fundamental domain for the group corresponding to the Farey symbol FS.¤k¼kö=«àáûÿûÿûÿûÿè÷à÷ îÔ÷oø= îÜ÷ûÿÿÿsù=
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	__sig__32$ŒhŒ™=¬å`öûÿÿÿš=FThe orthogonal complement of M, with respect to the monodromy pairing.H’›=´´ûÿûÿûÿûÿtö¬åhöD–œ=
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w? 126*x^16 - 298*x^17 - 396*x^18 - 391*x^19 - 269*x^20 - 54*x^21 + 172*x^22 + tûÓ? 304*x^23 + 285*x^24 + 135*x^25 - 84*x^26 - 317*x^27 - 525*x^28 - 662*x^29 - Ôù"?Àù,L?ôúüðúdú,ùI?(ý@,ý0ýÀù4ýDòJ?Ì0ˆˆTˆ$Àˆtˆ€‰‰œˆˆüˆÄŠÐŠŠ‹	‹à@Š	¸ì &0%L`Ž> pŽ=Ž1ŒŽ˜Ž¨Ž´ŽÀŽÐŽ!àŽŽÿÿÿÿŽ€ŽTüŽÿÿÿÿˆŽ0lx"		`¤ˆˆˆ˜Ø‰h¨°ÿÿÿÿô‘‘	‘èK?%c|c/@ 679*x^30 - 577*x^31 - 431*x^32 - 335*x^33 - 327*x^34 - 378*x^35 - 440*x^36 - ‹@ 484*x^37 - 486*x^38 - 415*x^39 - 263*x^40 - 77*x^41 + 64*x^42 + 105*x^43 +  ç@ 42*x^44 - 110*x^45 - 339*x^46 - 604*x^47 - 806*x^48 - 840*x^49 - 683*x^50 - ÐûCA 422*x^51 - 181*x^52 - 36*x^53 - 5*x^54 - 99*x^55 - 327*x^56 - 646*x^57 - - LŸA 944*x^58 - 1112*x^59 - 1111*x^60 - 972*x^61 - 750*x^62 - 503*x^63 - 7 - îA 300*x^64 - 220*x^65 - 287*x^66 - 425*x^67 - 509*x^68 - 475*x^69 - 355*x^70 - ŒýJB 227*x^71 - 148*x^72 - 158*x^73 - 300*x^74 - 599*x^75 - 996*x^76 - ™B 1344*x^77 - 1485*x^78 - 1353*x^79 - 990*x^80 - 513*x^81 - 71*x^82 +  ûèB 179*x^83 + 125*x^84 - 248*x^85 - 825*x^86 - 1413*x^87 - 1819*x^88 - ¸ü7C 1919*x^89 - 1671*x^90 - 1134*x^91 - 471*x^92 + 84*x^93 + 347*x^94 + „†C 264*x^95 - 82*x^96 - 539*x^97 - 944*x^98 - 1182*x^99 - 1199*x^100 - ÕC 1033*x^101 - 798*x^102 - 635*x^103 - 631*x^104 - 807*x^105 - 1113*x^106 - $D 1447*x^107 - 1655*x^108 - 1604*x^109 - 1259*x^110 - 742*x^111 - 265*x^112 - ðsD 27*x^113 - 113*x^114 - 512*x^115 - 1122*x^116 - 1765*x^117 - 2207*x^118 -  |ÂD 2270*x^119 - 1900*x^120 - 1206*x^121 - 390*x^122 + 304*x^123 + 664*x^124 + ÿlÿE 550*x^125 - 14*x^126 - 837*x^127 - 1622*x^128 - 2141*x^129 - 2294*x^130 - ÿøÿ`E 2100*x^131 - 1629*x^132 - 1029*x^133 - 498*x^134 - 240*x^135 - 318*x^136 - ä¯E 632*x^137 - 982*x^138 - 1224*x^139 - 1289*x^140 - 1193*x^141 - 995*x^142 - tþE 832*x^143 - 839*x^144 - 1079*x^145 - 1453*x^146 - 1786*x^147 - 1914*x^148 - ÔMF 1790*x^149 - 1413*x^150 - 837*x^151 - 189*x^152 + 274*x^153 + 319*x^154 -  œF 134*x^155 - 911*x^156 - 1734*x^157 - 2339*x^158 - 2588*x^159 - 2411*x^160 - XëF 1847*x^161 - 1039*x^162 - 270*x^163 + 211*x^164 + 271*x^165 - 42*x^166 - - è:G 605*x^167 - 1223*x^168 - 1709*x^169 - 1879*x^170 - 1710*x^171 - 1343*x^172 - t‰G 1033*x^173 - 917*x^174 - 1001*x^175 - 1173*x^176 - 1352*x^177 - 1436*x^178 - ØG 1347*x^179 - 1013*x^180 - 513*x^181 - 38*x^182 + 147*x^183 - 76*x^184 - 8 - <	'H 695*x^185 - 1538*x^186 - 2378*x^187 - 2911*x^188 - 2926*x^189 - 2347*x^190 - œvH 1359*x^191 - 254*x^192 + 631*x^193 + 1077*x^194 + 931*x^195 + 217*x^196 - - $ÅH 897*x^197 - 2031*x^198 - 2826*x^199 - 3035*x^200 - 2680*x^201 - 1921*x^202 - ´I 1043*x^203 - 275*x^204 + 165*x^205 + 216*x^206 - 85*x^207 - 548*x^208 - 2 - @cI 1014*x^209 - 1320*x^210 - 1426*x^211 - 1330*x^212 - 1184*x^213 - 208 - ¥I 1131*x^214 - 1294*x^215 - 1595*x^216 - 1888*x^217 - 1956*x^218 - t€ÿ€€€@B€@´€@L@Ñ@‚@Ñ@F@€@3€Î€à€G€~€@*@Œ@‡@@6€@¬€0‡€T€@=@‚@–‚@§‚@A‚@¯@O@G@z@¸@ä@æ@Ÿ@@M€@@€i€*€n€@S@\‚@&ƒ@Hƒ@«‚@¦@µ€@$€@€@c€@G@†‚@°ƒ@X„@W„@Ìƒ@î‚@÷@,@Ü€@@©@ý@Û@c@ã€@”€@ž€@,@W‚@äƒ@@…@Í…@I…@Þƒ@‚@G€@³€}€ø€@9ƒ@……@‡@‡@‡†@n„@×@T€[R€@‚@°ƒ@ž„@¯„@	„@ƒ@{‚@w‚@'ƒ@Y„@§…@w†@D†@ë„@æ‚@	@€@q€@‚@b„@å†@Ÿˆ@Þˆ@l‡@¶„@†@0˜‚&‚€@Eƒ@V†@]ˆ@öˆ@4ˆ@]†@„@ò@ð€@>@x‚@Öƒ@È„@	…@©„@ãƒ@@ƒ@Gƒ@7„@­…@ú†@z‡@þ†@……@Eƒ@½€@?†€@ƒ@Æ†@#‰@Š@k‰@7‡@„@@Ó€*€@]‚@Ç„@­†@W‡@®†@?…@	„@•ƒ@éƒ@•„@H…@œ…@C…@õƒ@‚@&€@“€L€@·‚@†@J‰@_‹@n‹@+‰@O…@þ€@w‚5„£ƒÙ€ƒ@ï‡@
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‹@ï‡@ƒ@Ù€£ƒ5„w‚þ€@O…@+‰@n‹@_‹@J‰@†@·‚@L€@“€&€@‚@õƒ@C…@œ…@H…@•„@éƒ@•ƒ@	„@?…@®†@W‡@­†@Ç„@]‚@*€@Ó€@„@7‡@k‰@Š@#‰@Æ†@ƒ@†€@?½€@Eƒ@……@þ†@z‡@ú†@­…@7„@Gƒ@@ƒ@ãƒ@©„@	…@È„@Öƒ@x‚@>@ð€@ò@„@]†@4ˆ@öˆ@]ˆ@V†@Eƒ@€@&‚˜‚0†@¶„@l‡@Þˆ@Ÿˆ@å†@b„@‚@q€@€@	@æ‚@ë„@D†@w†@§…@Y„@'ƒ@w‚@{‚@ƒ@	„@¯„@ž„@°ƒ@‚@R€@[T€×@n„@‡†@‡@‡@……@9ƒ@ø€@}€³€G€@‚@Þƒ@I…@Í…@@…@äƒ@W‚@,@ž€@”€@ã€@c@Û@ý@©@@Ü€@,@÷@î‚@Ìƒ@W„@X„@°ƒ@†‚@G@c€@€@$€@µ€@¦@«‚@Hƒ@&ƒ@\‚@S@n€@*€i€@€M€@@Ÿ@æ@ä@¸@z@G@O@¯@A‚@§‚@–‚@‚@=@T€@‡€0¬€6€@@‡@Œ@*@~€@G€à€Î€3€€@F@Ñ@‚@Ñ@L@´€@B€@€@€€çI 1737*x^219 - 1226*x^220 - 559*x^221 + 111*x^222 + 528*x^223 + 538*x^224 + 6J 51*x^225 - 798*x^226 - 1812*x^227 - 2675*x^228 - 3145*x^229 - 3006*x^230 - $d$…J 2298*x^231 - 1223*x^232 - 184*x^233 + 549*x^234 + 808*x^235 + 629*x^236 + $ð$ÔJ 54*x^237 - 729*x^238 - 1527*x^239 - 2042*x^240 - 2158*x^241 - 1887*x^242 - #K 1461*x^243 - 1038*x^244 - 779*x^245 - 681*x^246 - 747*x^247 - 818*x^248 - ¤rK 792*x^249 - 569*x^250 - 279*x^251 - 54*x^252 - 82*x^253 - 362*x^254 -  - ÁK 924*x^255 - 1675*x^256 - 2517*x^257 - 3134*x^258 - 3279*x^259 - 2758*x^260 - DüL 1729*x^261 - 448*x^262 + 682*x^263 + 1410*x^264 + 1490*x^265 + 930*x^266 -  ¤ý_L 166*x^267 - 1410*x^268 - 2474*x^269 - 3017*x^270 - 2952*x^271 - 2255*x^272 - @®L 1192*x^273 - 57*x^274 + 710*x^275 + 921*x^276 + 534*x^277 - 169*x^278 - 2 - \'ýL 972*x^279 - 1608*x^280 - 2019*x^281 - 2097*x^282 - 1942*x^283 - 1629*x^284 - LM 1410*x^285 - 1368*x^286 - 1566*x^287 - 1781*x^288 - 1867*x^289 - 2ŽM 1599*x^290 - 1010*x^291 - 168*x^292 + 600*x^293 + 1076*x^294 + 977*x^295 + ˜ÝM 331*x^296 - 774*x^297 - 1973*x^298 - 3002*x^299 - 3497*x^300 - 3353*x^301 - $
,N 2495*x^302 - 1223*x^303 + 126*x^304 + 1016*x^305 + 1203*x^306 + 576*x^307 - ð{N 544*x^308 - 1863*x^309 - 2937*x^310 - 3576*x^311 - 3565*x^312 - 3009*x^313 - 1ÊN 2022*x^314 - 1011*x^315 - 216*x^316 + 86*x^317 - 11*x^318 - 423*x^319 - 3 - ø
O 833*x^320 - 1126*x^321 - 1119*x^322 - 960*x^323 - 728*x^324 - 712*x^325 - hO 951*x^326 - 1523*x^327 - 2195*x^328 - 2855*x^329 - 3170*x^330 - 3032*x^331 - Ø·O 2285*x^332 - 1194*x^333 - 40*x^334 + 640*x^335 + 650*x^336 - 133*x^337 -  - dP 1368*x^338 - 2775*x^339 - 3839*x^340 - 4307*x^341 - 3893*x^342 - ^337 - HP 2779*x^343 - 1185*x^344 + 297*x^345 + 1303*x^346 + 1437*x^347 + 787*x^348 - €—P 497*x^349 - 1841*x^350 - 2878*x^351 - 3205*x^352 - 2947*x^353 - 2251*x^354 - æP 1576*x^355 - 1095*x^356 - 1089*x^357 - 1454*x^358 - 2162*x^359 - 5ÆQ 3959*x^372 - 2752*x^373 - 1064*x^374 + 471*x^375 + 1471*x^376 + 1439*x^377 + tR 450*x^378 - 1329*x^379 - 3169*x^380 - 4576*x^381 - 5022*x^382 - 4606*x^383 - ”$dR 3436*x^384 - 2061*x^385 - 782*x^386 - 127*x^387 - 166*x^388 - 977*x^389 - - ´9(Q 2783*x^360 - 3067*x^361 - 2683*x^362 - 1842*x^363 - 771*x^364 - 39*x^365 + wQ 142*x^366 - 457*x^367 - 1562*x^368 - 2942*x^369 - 4003*x^370 - 4454*x^371 - ³R 2060*x^390 - 3062*x^391 - 3437*x^392 - 3218*x^393 - 2431*x^394 - x^389 - õR 1600*x^395 - 993*x^396 - 1038*x^397 - 1605*x^398 - 2614*x^399 - 3506*x^400 - ø'DS 4035*x^401 - 3804*x^402 - 2961*x^403 - 1575*x^404 - 227*x^405 + 723*x^406 + €(“S 679*x^407 - 318*x^408 - 2134*x^409 - 3992*x^410 - 5387*x^411 - 5720*x^412 - P=âS 5094*x^413 - 3601*x^414 - 1891*x^415 - 359*x^416 + 349*x^417 + 167*x^418 - 01T 968*x^419 - 2447*x^420 - 3865*x^421 - 4563*x^422 - 4503*x^423 - 3624*x^424 - L€T 2493*x^425 - 1449*x^426 - 1084*x^427 - 1331*x^428 - 2169*x^429 - 2`U 5934*x^442 - 5099*x^443 - 3385*x^444 - 1505*x^445 + 97*x^446 + 708*x^447 + t¯U 330*x^448 - 1077*x^449 - 2859*x^450 - 4607*x^451 - 5545*x^452 - 5527*x^453 - ”$þU 4396*x^454 - 2824*x^455 - 1277*x^456 - 508*x^457 - 515*x^458 - 1371*x^459 - ´9MV 2520*x^460 - 3724*x^461 - 4340*x^462 - 4309*x^463 - 3438*x^464 - 1ÂT 3023*x^430 - 3674*x^431 - 3680*x^432 - 3157*x^433 - 2058*x^434 - 943*x^435 - U 146*x^436 - 293*x^437 - 1296*x^438 - 3017*x^439 - 4665*x^440 - 5831*x^441 - V 2301*x^465 - 1257*x^466 - 1019*x^467 - 1567*x^468 - 2936*x^469 - ÑV 4415*x^470 - 5616*x^471 - 5836*x^472 - 5116*x^473 - 3448*x^474 - W 1582*x^475 + 61*x^476 + 720*x^477 + 362*x^478 - 1139*x^479 - 3114*x^480 - bW 5107*x^481 - 6210*x^482 - 6247*x^483 - 5028*x^484 - 3264*x^485 - x^480 - ¤W 1431*x^486 - 373*x^487 - 164*x^488 - 989*x^489 - 2283*x^490 - 3795*x^491 - $óW 4787*x^492 - 5137*x^493 - 4536*x^494 - 3482*x^495 - 2243*x^496 - *
X¦7¥‘³FlÑ Ïb>t5X 1584*x^497 - 1564*x^498 - 2391*x^499 - 3477*x^500 - 4536*x^501 - wX 4849*x^502 - 4442*x^503 - 3201*x^504 - 1804*x^505 - 544*x^506 - 106*x^507 - ´@ÆX 469*x^508 - 1840*x^509 - 3626*x^510 - 5472*x^511 - 6453*x^512 - 6381*x^513 - @AY 5018*x^514 - 3082*x^515 - 1020*x^516 + 282*x^517 + 705*x^518 - 72*x^519 - - VdY 1529*x^520 - 3390*x^521 - 4778*x^522 - 5471*x^523 - 5054*x^524 - x^519 - ¦Y 4001*x^525 - 2524*x^526 - 1443*x^527 - 947*x^528 - 1457*x^529 - 2491*x^530 - ¬+õY 3791*x^531 - 4522*x^532 - 4618*x^533 - 3835*x^534 - 2745*x^535 - 97Z 1524*x^536 - 873*x^537 - 848*x^538 - 1813*x^539 - 3238*x^540 - 4809*x^541 - tR†Z 5588*x^542 - 5410*x^543 - 4008*x^544 - 2082*x^545 - 25*x^546 + 1227*x^547 + üRÕZ 1511*x^548 + 468*x^549 - 1330*x^550 - 3558*x^551 - 5260*x^552 - 6191*x^553 - Èþ$[ 5844*x^554 - 4628*x^555 - 2734*x^556 - 1097*x^557 - 52*x^558 - 191*x^559 -  äs[ 1096*x^560 - 2558*x^561 - 3715*x^562 - 4423*x^563 - 4240*x^564 - *µ[ 3541*x^565 - 2346*x^566 - 1379*x^567 - 789*x^568 - 1103*x^569 - 1937*x^570 - ÀO\ 3153*x^571 - 3931*x^572 - 4143*x^573 - 3361*x^574 - 2074*x^575 - 478*x^576 + LPS\ 548*x^577 + 818*x^578 - 111*x^579 - 1725*x^580 - 3808*x^581 - 5456*x^582 -  eMd 25*x^732 - 2082*x^733 - 4008*x^734 - 5410*x^735 - 5588*x^736 - 4809*x^737 - Øœd 3238*x^738 - 1813*x^739 - 848*x^740 - 873*x^741 - 1524*x^742 - 2745*x^743 - ø ëd 3835*x^744 - 4618*x^745 - 4522*x^746 - 3791*x^747 - 2491*x^748 - 5-e 1457*x^749 - 947*x^750 - 1443*x^751 - 2524*x^752 - 4001*x^753 - 5054*x^754 - Ô\|e 5471*x^755 - 4778*x^756 - 3390*x^757 - 1529*x^758 - 72*x^759 + 705*x^760 +  `]Ëe 282*x^761 - 1020*x^762 - 3082*x^763 - 5018*x^764 - 6381*x^765 - 6453*x^766 - 0rf 5472*x^767 - 3626*x^768 - 1840*x^769 - 469*x^770 - 106*x^771 - 544*x^772 -  T‡if 1804*x^773 - 3201*x^774 - 4442*x^775 - 4849*x^776 - 4536*x^777 - *«f 3477*x^778 - 2391*x^779 - 1564*x^780 - 1584*x^781 - 2243*x^782 - íf 3482*x^783 - 4536*x^784 - 5137*x^785 - 4787*x^786 - 3795*x^787 - /g 2283*x^788 - 989*x^789 - 164*x^790 - 373*x^791 - 1431*x^792 - 3264*x^793 - QÀQ~g 5028*x^794 - 6247*x^795 - 6210*x^796 - 5107*x^797 - 3114*x^798 - *Àg 1139*x^799 + 362*x^800 + 720*x^801 + 61*x^802 - 1582*x^803 - 3448*x^804 - h 5116*x^805 - 5836*x^806 - 5616*x^807 - 4415*x^808 - 2936*x^809 - x^804 - Qh 1567*x^810 - 1019*x^811 - 1257*x^812 - 2301*x^813 - 3438*x^814 - “h 4309*x^815 - 4340*x^816 - 3724*x^817 - 2520*x^818 - 1371*x^819 - 515*x^820 - Œâh 508*x^821 - 1277*x^822 - 2824*x^823 - 4396*x^824 - 5527*x^825 - 5545*x^826 - 1i 4607*x^827 - 2859*x^828 - 1077*x^829 + 330*x^830 + 708*x^831 + 97*x^832 - - è%€i 1505*x^833 - 3385*x^834 - 5099*x^835 - 5934*x^836 - 5831*x^837 - x^832 - Âi 4665*x^838 - 3017*x^839 - 1296*x^840 - 293*x^841 - 146*x^842 - 943*x^843 - a¸aj 2058*x^844 - 3157*x^845 - 3680*x^846 - 3674*x^847 - 3023*x^848 - *Sj 2169*x^849 - 1331*x^850 - 1084*x^851 - 1449*x^852 - 2493*x^853 - •j 3624*x^854 - 4503*x^855 - 4563*x^856 - 3865*x^857 - 2447*x^858 - 968*x^859 + Pšäj 167*x^860 + 349*x^861 - 359*x^862 - 1891*x^863 - 3601*x^864 - 5094*x^865 -  àš3k 5720*x^866 - 5387*x^867 - 3992*x^868 - 2134*x^869 - 318*x^870 + 679*x^871 + ˜‚k 723*x^872 - 227*x^873 - 1575*x^874 - 2961*x^875 - 3804*x^876 - 4035*x^877 - ¸Ñk 3506*x^878 - 2614*x^879 - 1605*x^880 - 1038*x^881 - 993*x^882 - 1600*x^883 - Ô0 l 2431*x^884 - 3218*x^885 - 3437*x^886 - 3062*x^887 - 2060*x^888 - 977*x^889 - øEol 166*x^890 - 127*x^891 - 782*x^892 - 2061*x^893 - 3436*x^894 - 4606*x^895 -  [¾l 5022*x^896 - 4576*x^897 - 3169*x^898 - 1329*x^899 + 450*x^900 + 1439*x^901 + <pm 1471*x^902 + 471*x^903 - 1064*x^904 - 2752*x^905 - 3959*x^906 - 4454*x^907 - `…\m 4003*x^908 - 2942*x^909 - 1562*x^910 - 457*x^911 + 142*x^912 - 39*x^913 - - €š!
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„½«m 771*x^914 - 1842*x^915 - 2683*x^916 - 3067*x^917 - 2783*x^918 - 2162*x^919 - úm 1454*x^920 - 1089*x^921 - 1095*x^922 - 1576*x^923 - 2251*x^924 - 6<n 2947*x^925 - 3205*x^926 - 2878*x^927 - 1841*x^928 - 497*x^929 + 787*x^930 + Ô7‹n 1437*x^931 + 1303*x^932 + 297*x^933 - 1185*x^934 - 2779*x^935 - 3893*x^936 - `8Ún 4307*x^937 - 3839*x^938 - 2775*x^939 - 1368*x^940 - 133*x^941 + 650*x^942 + 0M)o 640*x^943 - 40*x^944 - 1194*x^945 - 2285*x^946 - 3032*x^947 - 3170*x^948 - Tbxo 2855*x^949 - 2195*x^950 - 1523*x^951 - 951*x^952 - 712*x^953 - 728*x^954 - wxwÇo 960*x^955 - 1119*x^956 - 1126*x^957 - 833*x^958 - 423*x^959 - 11*x^960 +  Œ˜Œ´p 1016*x^973 + 126*x^974 - 1223*x^975 - 2495*x^976 - 3353*x^977 - 3497*x^978 - Dq 3002*x^979 - 1973*x^980 - 774*x^981 + 331*x^982 + 977*x^983 + 1076*x^984 +  d$Rq 600*x^985 - 168*x^986 - 1010*x^987 - 1599*x^988 - 1867*x^989 - 1781*x^990 - „9¡q 1566*x^991 - 1368*x^992 - 1410*x^993 - 1629*x^994 - 1942*x^995 - 1ãq 2097*x^996 - 2019*x^997 - 1608*x^998 - 972*x^999 - 169*x^1000 + 534*x^1001 + hu2r 921*x^1002 + 710*x^1003 - 57*x^1004 - 1192*x^1005 - 2255*x^1006 - tr 2952*x^1007 - 3017*x^1008 - 2474*x^1009 - 1410*x^1010 - 166*x^1011 + Tœ¶r 930*x^1012 + 1490*x^1013 + 1410*x^1014 + 682*x^1015 - 448*x^1016 -  Øœør 1729*x^1017 - 2758*x^1018 - 3279*x^1019 - 3134*x^1020 - 2517*x^1021 - 0®p 86*x^961 - 216*x^962 - 1011*x^963 - 2022*x^964 - 3009*x^965 - 3565*x^966 - ep 3576*x^967 - 2937*x^968 - 1863*x^969 - 544*x^970 + 576*x^971 + 1203*x^972 + :s 1675*x^1022 - 924*x^1023 - 362*x^1024 - 82*x^1025 - 54*x^1026 - 279*x^1027 - °	‰s 569*x^1028 - 792*x^1029 - 818*x^1030 - 747*x^1031 - 681*x^1032 - 9Ës 779*x^1033 - 1038*x^1034 - 1461*x^1035 - 1887*x^1036 - 2158*x^1037 - l0t 2042*x^1038 - 1527*x^1039 - 729*x^1040 + 54*x^1041 + 629*x^1042 + - ì0Ot 808*x^1043 + 549*x^1044 - 184*x^1045 - 1223*x^1046 - 2298*x^1047 - SÜS‘t 3006*x^1048 - 3145*x^1049 - 2675*x^1050 - 1812*x^1051 - 798*x^1052 + `TÓt 51*x^1053 + 538*x^1054 + 528*x^1055 + 111*x^1056 - 559*x^1057 - 2 + ¸eu 1226*x^1058 - 1737*x^1059 - 1956*x^1060 - 1888*x^1061 - 1595*x^1062 - üˆWu 1294*x^1063 - 1131*x^1064 - 1184*x^1065 - 1330*x^1066 - 1426*x^1067 - |‰™u 1320*x^1068 - 1014*x^1069 - 548*x^1070 - 85*x^1071 + 216*x^1072 +  - ÔšÛu 165*x^1073 - 275*x^1074 - 1043*x^1075 - 1921*x^1076 - 2680*x^1077 - ¾v 3035*x^1078 - 2826*x^1079 - 2031*x^1080 - 897*x^1081 + 217*x^1082 + ˜¾¡v 2347*x^1088 - 2926*x^1089 - 2911*x^1090 - 2378*x^1091 - 1538*x^1092 - 4	ãv 695*x^1093 - 76*x^1094 + 147*x^1095 - 38*x^1096 - 513*x^1097 - 1013*x^1098 - ˆ,2w 1347*x^1099 - 1436*x^1100 - 1352*x^1101 - 1173*x^1102 - 1001*x^1103 - 098 - -tw 917*x^1104 - 1033*x^1105 - 1343*x^1106 - 1710*x^1107 - 1879*x^1108 - ¨A¶w 1709*x^1109 - 1223*x^1110 - 605*x^1111 - 42*x^1112 + 271*x^1113 + - PSøw 211*x^1114 - 270*x^1115 - 1039*x^1116 - 1847*x^1117 - 2411*x^1118 - ˜v:x 2588*x^1119 - 2339*x^1120 - 1734*x^1121 - 911*x^1122 - 134*x^1123 + w|x 319*x^1124 + 274*x^1125 - 189*x^1126 - 837*x^1127 - 1413*x^1128 -  lˆ¾x 1790*x^1129 - 1914*x^1130 - 1786*x^1131 - 1453*x^1132 - 1079*x^1133 - ¸«y 839*x^1134 - 832*x^1135 - 995*x^1136 - 1193*x^1137 - 1289*x^1138 - - 8¬By 1224*x^1139 - 982*x^1140 - 632*x^1141 - 318*x^1142 - 240*x^1143 - ½ˆ½„y 498*x^1144 - 1029*x^1145 - 1629*x^1146 - 2100*x^1147 - 2294*x^1148 - Æy 2141*x^1149 - 1622*x^1150 - 837*x^1151 - 14*x^1152 + 550*x^1153 + -  z 664*x^1154 + 304*x^1155 - 390*x^1156 - 1206*x^1157 - 1900*x^1158 - &ˆ&Jz 2270*x^1159 - 2207*x^1160 - 1765*x^1161 - 1122*x^1162 - 512*x^1163 - 'Œz 113*x^1164 - 27*x^1165 - 265*x^1166 - 742*x^1167 - 1259*x^1168 -  - `8Îz 1604*x^1169 - 1655*x^1170 - 1447*x^1171 - 1113*x^1172 - 807*x^1173 - ¤[{ 631*x^1174 - 635*x^1175 - 798*x^1176 - 1033*x^1177 - 1199*x^1178 -  $\R{ 1182*x^1179 - 944*x^1180 - 539*x^1181 - 82*x^1182 + 264*x^1183 +  m|m”{ 347*x^1184 + 84*x^1185 - 471*x^1186 - 1134*x^1187 - 1671*x^1188 - Ö{ 1919*x^1189 - 1819*x^1190 - 1413*x^1191 - 825*x^1192 - 248*x^1193 + `¢| 125*x^1194 + 179*x^1195 - 71*x^1196 - 513*x^1197 - 990*x^1198 -  + à¢Z| 1353*x^1199 - 1485*x^1200 - 1344*x^1201 - 996*x^1202 - 599*x^1203 - ÐÅœ| 300*x^1204 - 158*x^1205 - 148*x^1206 - 227*x^1207 - 355*x^1208 - - PÆÞ| 475*x^1209 - 509*x^1210 - 425*x^1211 - 287*x^1212 - 220*x^1213 - _v 931*x^1083 + 1077*x^1084 + 631*x^1085 - 254*x^1086 - 1359*x^1087 -  } 300*x^1214 - 503*x^1215 - 750*x^1216 - 972*x^1217 - 1111*x^1218 - b} 1112*x^1219 - 944*x^1220 - 646*x^1221 - 327*x^1222 - 99*x^1223 - 5*x^1224 - .±} 36*x^1225 - 181*x^1226 - 422*x^1227 - 683*x^1228 - 840*x^1229 - 806*x^1230 - Œ.~ 604*x^1231 - 339*x^1232 - 110*x^1233 + 42*x^1234 + 105*x^1235 + 64*x^1236 - XCO~ 77*x^1237 - 263*x^1238 - 415*x^1239 - 486*x^1240 - 484*x^1241 - 440*x^1242 - |Xž~ 378*x^1243 - 327*x^1244 - 335*x^1245 - 431*x^1246 - 577*x^1247 - 0k€ x^1278; à~ 679*x^1248 - 662*x^1249 - 525*x^1250 - 317*x^1251 - 84*x^1252 + 135*x^1253 + \”/ 285*x^1254 + 304*x^1255 + 172*x^1256 - 54*x^1257 - 269*x^1258 - 391*x^1259 - ì”~ 396*x^1260 - 298*x^1261 - 126*x^1262 + 71*x^1263 + 224*x^1264 + 274*x^1265 + À©Í 206*x^1266 + 51*x^1267 - 141*x^1268 - 326*x^1269 - 465*x^1270 - 518*x^1271 - Ü¾
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