########################### # examP # for MapleV Releases 3 to 5.1 (at least) ############################################## Exam:=proc() local a,a1,a2,a3,a4,a6,b,c,DD,l,t,c4,c6;global i,iee; l:=proc();lprint(args);end; if nargs=0 then l(` Exam(args); will print examples according to the following menu:`): exam(); l(``); elif nargs=1 then i:=args: if not type(i,posint) or i>6 then print(qts):RETURN();fi; if i=1 then l(`POSSIBLE TORSION subgroups of E(Q):`); l(` Mazur's famous theorem is that the possibile torsion subgroups of\ E(Q)`); l(` -- here denoted T -- are, where Cn denotes the cyclic group of order\ n:`); print(`Cn for 1<=n<=10 and n=12, and`); print(`C2 X Cn for n=2,4,6,8.`); l(` The following tables give parametrizations (all genus 0 from the`); l(` modular curve X_1 point of view), originally due to Tate (?), of all`); l(` curves/Q with at least the indicated T; that is, any choice of the`); l(` parameter(s) subject to the condition that the discriminant DD<>0`); l(` gives an E with at least that torsion. For each possible group we`); l(` give the curve with smallest conductor Nc whose T is exactly that g\ roup,`); l(` and sometimes a simpler example with larger Nc. All these examples a\ re`); l(` in the starter catalog of apecs, and all happen to have rank r4=0.`); l(` `); l(` The E with trivial torsion and smallest Nc is C11.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` The cases C2,C3,C2XC2 don't fit well into the table so we list`); l(` them separately:`); print(`C2: `,Y^2+a1*X*Y=X^3+a2*X^2+a4*X); print(` such that DD`=a4^2*(a1^4+8*a1^2*a2+16*a2^2-64*a4),`<>0`); l(`or, completing the square, Y^2=X^3+a2 X^2 +a4 X with DD=16a4^2(a2^2-4\ a4)<>0.`); l(` Example A64: `,Y^2=X^3+X,` ; the example with smallest conductor is`); l(` E14: `,Y^2+X*Y+Y=X^3-171*X-874,` with point `,[X,Y]=[15,-8]); l(` of order 2. One can replace `,X,Y,` with `,X+15,Y-8); l(` to obtain an equation with a3=a6=0.`); l(` Also: E has a point of order 2 iff jay=(t+16)^3/t, some t in Q* ---`); l(` See Exam(3).`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); print(`C2XC2: `,Y^2=X^3+a2*X^2+(a2^2/4-a^2)*X,` a in Q*, DD=`,\ 4*a^2*(a2^2-4*a^2)^2); print(` Example A32: `,Y^2=X^3-X,` ; smallest Nc example: E15.`); l(`Command to generate examples: Exam(4,a2,a)`); print(`C3: `,Y^2+a1*X*Y+a3*Y=X^3); print(` such that DD`=a3^3*(a1^3-27*a3),`<>0`); print(` Example A27: `,Y^2+Y=X^3,` ; smallest Nc example: A19`); l(`Command to generate examples: Exam(3,a1,a3)`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` Next for Cn with n>3, the curve is`);exam(1): elif i=2 then l(` COMPLEX MULTIPLICATION:`); l(` An elliptic curve E/Q has CM when there is a (non-trivial)`); l(` twist E'/Q that is isogenous to E. Since there is a slight subtlety`); l(` when jay=1728, let us recall some generalities about twists.`); l(` Two elliptic curves defined over a field K are twists of one`); l(` another when they become isomorphic over an extension field. This is`); l(` so iff they have the same jay, and is therefore easily detected.`); l(` To avoid technicalities let us suppose that char(K)<>2 or 3.`); l(` Then E, and similarly E', can be taken in the Weierstrass "c-form"`); print(Y^2=X^3-c4/48*X-c6/864); l(` The transformation equations of an isomorphism E-->E' defined over`); l(` an extension of K (u*a1'=a1+2s, etc. in the usual notation) imply`); l(` r=s=t=0 and u^4*c4'=c4, u^6*c6'=c6. When c4<>0 (<==>jay=c4^3/DD<>0`); l(` where DD denotes the discriminant in apecs) and c6<>0 (<==>jay<>1728`); l(` since jay-1728=c6^2/DD) this implies that u^2 is in K and the twist`); l(` is quadratic. Writing u^2=a, E' has the c-form`); print(Y^2=X^3-a^2*c4/48*X-a^3*c6/864); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` We indicate such a quadratic twist by E'=E*a; then also E=E'*a, by an`); l(` obvious change of variables in the above equations.`); l(` When jay=1728, then E: Y^2=X^3-c4*X/48 is isomorhic to`); l(` E': Y^2=X^3-b*c4*X/48 only over Q(b^(1/4)), and so E' may be a`); l(` quartic twist. Similarly there are cubic and sextic twists when`); l(` jay=0. However for any curve with jay=0 or 1728 there always exists`); l(` a rationally defined isogeny to a quadratic twist (there is always a`); l(` "rational manifestation" of CM), as we now explain.`); l(` Let H be the subgroup of order 2 generated by the point [0,0] on`); l(` E: Y^2=X^3+BX. Then Velu's formulas give E/H=E': Y^2=X^3-4BX. This`); l(` is the only non-obvious quadratic twist: -4=(1+I)^4 and so E and E'`); l(` are isomorphic over K(I). Secondly, if H is the subgroup generated`); l(` by the point [0,sqrt(C)] of order 3 on E: Y^2=X^3+C, then H is`); l(` rationally defined and E/H=E': Y^2=X^3+(-3)^3C this time is a`); l(` "natural" quadratic twist.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` There are just 13 values of jay in Q that possess CM. We tabulate`); l(` these values along with the smallest conductor example E, and`); l(` d=the degree of the isogeny to E', with the standard apecs`); l(` convention: a minus sign is attached to d when the kernel is not`); l(` defined pointwise/Q. We also list D=the disriminant of the`); l(` endomorphism ring End_C(E), which is an order in an imaginary`); l(` quadratic field. We have E'=E*D for all our examples except the`); l(` pair for jay=1728. Any E''/Q with CM has jay in the list and E''`); l(` is a twist (possibly non-quadratic when jay=0 or 1728) of the`); l(` corresponding E in the list. When quadratic, say E''=E*a, then`); l(` E'' is isogenous to E'*a.`); l(` The equations of the isogeny E ---> E' and also those of the`); l(` dual E' ---> E have been worked out and stored in the catalog for`); l(` all d<=27. This means that for any E''/Q with any CM except (for`); l(` the time being) the three top values with CM by -43,-67 or -163,`); l(` the equations of the isogeny E'' ---> E''' and its dual can be`); l(` calculated immediately: the apecs command to have this`); l(` done, with the "present curve"=E'', is Tri(E'''). (If E''' is not`); l(` yet in the catalog then do Isog first). For the dual isogeny`); l(` change the present curve to E''' (by Ein or Go) and then Tri(E'').`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); exam(2); elif i=3 then l(` ISOGENY NETWORKS possible/Q:`); l(` The number of curves in an isogeny class/Q is 1,2,3,4,6 or 8`); l(` (Kenku, J. Num. Th., '82). Most of these numbers can occur in`); l(` distinct ways; we give examples of all possibilities below. As in`); l(` Antwerp IV, we draw a network: the vertices are the curves of the`); l(` class, the edges are the isogenies of prime degree and the weight`); l(` of an edge is the degree. Clearly every isogeny of composite degree`); l(` is the composition of isogenies of prime degree.`); l(` Following on his proof that the only possible values of`); l(` |tor(E(Q))| (denoted NN in apecs) are 1..10, 12, and 16, Mazur prove\ d`); l(` that the degrees of isogenies/Q with cyclic kernel that occur are:`); print(`1..19,21,25,27,37,43,67 and 163`); l(` When jay<>0 or 1728 the isogeny network that a particular E has,`); l(` and its position in that network (up to network isomorphism) are`); l(` entirely determined by the`); l(` value of jay because two E's with the same jay are quadratic twists`); l(` of one another and isogenies survive quadratic twists: if f:E-->E'`); l(` is an isogeny then an obvious definition gives f*a:E*a-->E'*a`); l(` (and the apecs procedures Isog and Tri take advantage of this`); l(` whenever possible). But for each of jay=0 and 1728, several differe\ nt`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` networks of isogenies occur; see below. Thus in a list of isogeny`); l(` network type vs. jay, a particular jay <>0,1728 will occur at most`); l(` once. One could refine this by making the network partially`); l(` directed: let E-->E' signify that all the points in the kernel of`); l(` the isogeny going in the direction of the arrow are rational.`); l(` Thus if p denotes the degree,`); l(` (i) we always have E<-->E' when p=2;`); l(` (ii) we never have E<-->E' when p>2 ? I don't know for sure;`); l(` (iii) (Serre) when E (hence also E') is semi-stable i.e. all`); l(` bad primes are multiplicative, we have either --> or <--.`); l(` Isog uses (i) and (iii).`); l(` I suppose there are moduli schemes that determine all jay in Q`); l(` attached to a particular vertex of one of these networks, but we`); l(` content ourselves with the usual modular curves X0(N) which`); l(` classify E with a cyclic isogeny of degree N. Let I(N) denote the`); l(` set of jay that occur and let g denote the genus of X0(N)`); l(` g=0<==>N is one of 1..10,12,13,16,18,25<==>there are infinitel\ y many`); l(` jay with a cyclic isogeny of degree N. Then I(N) is parametrized by\ the`); l(` N-th Fricke polynomial frN. Apecs knows only fr2,fr3,fr5,fr7,fr13,\ and`); l(` we only list these; we'll add the cases N=4,6,8,9,10,12,16,18,25 lat\ er.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` N || jay in I(N) parametrized by T in Q*`); l(` ----||----------------------------------------`); l(` 2 || (T + 16)^3/T`); l(` 3 || (T + 27)(T + 3)^3/T`); l(` 5 || (T^2 + 10T + 5)^3/T`); l(` 7 || (T^2 + 13T + 49)(T^2 + 5T + 1)^3/T`); l(` 13 || (T^2 + 5T + 13)(T^4 + 7T^3 + 20T^2 + 19T + 1)^3/T`); l(``); l(` fr||N=T*(R-jay), where R denotes the entry in the right column for N.`); l(` There's only one rational T for a given jay except when N=2 and`); l(` jay=1728. For an isogeny E---E' we have the pairs jay, jay' and T,T\ '.`); l(` In cases where E--E' arises from CM, T=T' and jay=jay'; see Exam(2)\ .`); l(` The relationship in the five cases is`); print(`TT'`=2^12,`TT'`=3^6,`TT'`=5^3,`TT'`=7^2,`TT'`=13); l(` This allows easy calculation of the "isogenous" jay' starting from j\ ay:`); l(` f:=op(1,fr||N):T:=roots(fr||N)[1][1]:jayprimed:=f;T:='T':`); l(` --must restore capital letters in apecs to "unevaluated".`); l(` We can use apecs to work out the case N=2 explicitly. We list, wi\ th`); l(` comments, the commands entered then the output sent to the screen.`); l(` 1. genj((T+16)^3/T); #Menu(genj) describes this command ===> `); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` [1, 0, 0,-36*T/(T**3+48*T**2-960*T+4096),-T/(T**3+48*T**2-960*T+4096)\ ]`); l(` This list, called "yo", = the Weier. coef.'s; to make it more readabl\ e`); l(` 2. factor(yo); (or simply factor(%);) ===>`); print([1, 0, 0, -36*T/(T+64)/(T-8)**2, -T/(T+64)/(T-8)**2]); l(` 3. x:='x':Q1:=absc(solve(prac,x)); ===> [2/(T-8),-1/(T-8)]`); l(` This is a point of order 2 on the curve--2/(T-8) is a root`); l(` of the apecs variable prac:=x^3+(b2/4)*x^2+(b4/2)*x+b6/4 (see Nota()\ ;)`); l(` solve ==> a list of 3 roots, absc picks the first one, disregarding th\ e`); l(` irrat. roots and ==> a point with the given abscissa 2/(T-8).`); l(` 4. velu(Q1);factor(jay); ===> (T + 256)^3/T^2`); l(` the "present" curve is now the 2-isogenous curve E/

. To see its`); l(` Weierstrass coef.'s in a readable form do factor(we()); (see Menu(we\ )).`); l(` 5. solve((T1+16)^3/T1)=jay,T1)[1]; ===> 4096/T`); l(` T1 stands for T'. The two equality cases T=T'=-64 and T=T'=64 corres\ pond`); l(` to CM by -4: A32--B32 and CM by -8: A256--B256.`); l(` The following is a complete list of finite non-empty I(N). A CM`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` isogeny of degree N contributes one jay to I(N), while an ordinary`); l(` isogeny contributes two. See Exam(2) for more info. on the CM cases.`); l(` The remaining cases N=4,6,8,9,10,12,16,18,25 have g=0 and infinite I\ (N).`); l(` The jay for N=19 is misprinted in Antwerp IV=yellow lectures #476, \ p.79.`); l(` N || CM jay || non-CM pair jay,jay' || example`); l(`-----||--------------||------------------------------||--------------`); l(` 11 || -2^(15) || || D121--E121`); l(` 11 || || -11^2 , -11*131^3 || F121--G121`); l(` 14 || || -15^3 , (3*5*17)^3 || A49--D49`); l(` 15 || || -5*29^3/2^5,-5^2*241^3/8 || A50--D50`); l(` " || || 5*211^3/2^(15),-25/2 || B50--C50`); l(` 17 || ||-17^2*101^3/2,-17*373^3/2^(17)|| A--B14450`); l(` 19 || -2^(15)*3^3 || || A--B361`); l(` 21 || || 15^3/2 , -9*(5*101/2^7)^3 || A--C162`); l(` " || || -9*(25/2)^3,-(3*5*383)^3/2^7 || B--D162`); l(` 27 || -3*(32*5)^3 || || C--D27`); l(` 37 || || -7*11^3,-7*137^3*2083^3 || A--B1225`); l(` 43 || -(64*3*5)^3 || || A--B1849`); l(` 67 ||-(32*3*5*11)^3|| || A--B4489`); l(` 163 ||-(64*3*5*23*29)^3 || || A--B26569`); l(` Finally we list all the (undirected) networks that occur.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); exam(3); elif i=4 then l(`EXAMPLES/Q of (succesively) larger rank`); l(` Let E be an elliptic curve defined over Z and let E(Q)=T x Z^r`); l(`where T is the finite torsion subgroup. The apecs starter catalog, a\ s of`); l(`version 6, contains all the examples in the table below; ncur=the (apecs`); l(`starter) catalog number. To call up the curve from the cat. use Ein(ncur).`); l(`Alternatively, when there is an entry in the args column in the table,`); l(`Exam(args) will produce that curve; in fact many of these curves`); l(`were discovered by systematically taking the parameter(s) in args in the`); l(`sequence of rationals ordered according to naive height. For instance`); l(` `); print(`exam(16,1,5/12);rkNC();`); l(`turned up the example with T = C2xC8, r = 3; this curve was found in a`); l(`different way by Andrej Dujella (internet posting Apr. 26/00). For`); l(`details see Menu(Ein0). Also, the example T = C2xC8, r = 2 was found in`); l(`a different way by Randall Rathbun (internet Sep. 25/99); for details`); l(`see Menu(Gcub). For a list of all E/Q up to conductor 5077, see`); l(`Cremona's web-published tables:`); print(`http://www.maths.nottingham.ac.uk/personal/jec`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` New in apecs ver. 5.7: there are better examples and new examples`); l(`in the table, including many from Tom Womack; see`); l(` http://tom.womack.net/maths/conductors.htm`); l(` " " " /torstab.htm`); l(` " " /mordellc.htm (for record y^2=X^3+k)`); l(`( " " /conductors.html is out of date)`); l(` New in apecs ver. 6: The example with T = C3, r = 4 from Garikai`); l(`Campbell's thesis:`); l(` http://www.swarthmore.edu/NatSci/gcampbel/`); l(`This example (which can be confirmed by the sequence of apecs commands`); print(`Exam(3,-109,421);RkNC();Seek(1200);)`); l(`inspired me to use the command Weier_fit to find the examples of T = C2`); l(`and r = 4,5 with relatively small conductor; see Menu(Weier_fit) for`); l(`details. For confimation of these two examples: RkNC().`); l(` Also first appearing this version: table entries for r = 3 and`); l(`T = C7, C8, C2xC6 (found by Exam(--,a) for a of incr. naive height).`); l(` `); l(` The examples above the dotted line have the smallest Nc for the given`); l(`T and r. Those below the line have the smallest Nc that I know of.`); l(`See Exam(1) for minimum Nc examples of r=0 and all possible T.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); exam(4); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(`The curves with T=O and r=4,5 are, respectively,`); l(` y^2+xy=x^3-x^2-79x+289,`); l(` y^2+y=x^3-79x+342;`); l(`the first was found with the aid of the apecs command Cubic_fit,`); l(`the 2nd is from Brumer & McGuiness, The behavior..., Bull. AMS,`); l(`23(1990), 375--382 (and rediscovered by Cubic_fit).`); l(`The examples of r=6 to 10 are all due to Tom Womack; several of them`); l(`replace earlier examples with larger conductors that are still in`); l(`the catalog: the example n=605 for r=6 improves previous entries in`); l(`the table obtainable from the cat. by Ein(n) for n=587, 586, and 539`); l(`(in order of increasing Nc).`); l(` The example of r=7 improves on Ein(548), also found by Womack, and`); l(`has a slightly smaller conductor than Ein(453), an example emailed to`); l(`me by Jim Buddenhagen.`); l(` The example of r=8 improves the close runner-up E(454) from Mestre,`); l(`Formules explicites..., Comp. Math. 58(1986); Ein(607) (Womack) lies`); l(`between these two.`); l(` The example of r=9 improves on Ein(550), another example from`); l(`Jim Buddenhagen, which in turn improves on Ein(455) from Mestre, ibid.`); l(` The example of r=10 improves on Mestre's Ein(456).`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` The examples of r=11..14 are all from Mestre's paper; but note that`); l(`for r=13 his D and abscissae x seem to have been miscopied from another`); l(`example.`); l(` A number of these curves have catalog entries indicating that apecs`); l(`commands such as Isog, RkNC, Rk1, Seek, Bas, ... have not been`); l(`carried out, in case one wants to see apecs at work. To verify the rank,`); l(`up to r=9 Rk1 works well. For r=10 and higher it is preferable to use`); l(`Mest and Seek1 alternately until the lower and upper bounds for the rank`); l(`coincide, the reason being that these curves have r independent integral`); l(`points --- Rk1 uses seek rather than seek1, which takes too long. And`); l(`for higher r eschew Rk which can get into interminable L-function`); l(`calculations.`); l(` For ncur=469..474 the catalog contains examples with T=C2 and r=7\ ..9`); l(`taken from Kretchmer, Construction of Elliptic Curves..., Math. Comp.`); l(`46(1986), 627--635.`); l(` Mestre and others have published examples of r=15,16,...; the big\ gest`); l(`as far as I know: an example with r>=24 due to Roland Martin and`); l(`William McMillen -- an internet posting Jan. 2000.`); l(`However apecs cannot reasonably handle such big examples; for one thing,`); l(`it bogs down trying to factor the discriminant DD --- but see`); l(`Menu(Afac) for a way around this problem.`); elif i=5 then l(` Here we seek "earliest" E/Q of certain types, earliest`); l(`meaning with the smallest conductor. Of course better examples are`); l(`always welcome: e-mail connell@math.mcgill.ca.`); l(` Most of the examples that follow became part of the apecs catalog`); l(`in Aug. 1997 and Feb. 1998. So these curves are in the catalog`); l(`for apecs version>= 4.36.`); l(` We will refer to seven papers:`); l(`[1] K. Kramer, A family of semistable elliptic curves with large Tate-`); l(` Shafarevitch groups, Proc. Amer. Math. Soc., 89(1983), 379--386.`); l(`[2] G. Kramarz and D. Zagier, Numerical investigations related to the`); l(`L-series of certain elliptic curves, J. Indian Math. Soc., 52(1987), 51\ --69`); l(`[3] J. Cremona, The analytic order of III for modular elliptic curves,`); l(` J. Th. Nombres Bordeaux 5(1993), 179--184.`); l(`[4] B.M.M. deWeger, A+B=C and big III's, Quar. J. Math. Oxford,`); l(`49(1998), 105--128.`); l(`[5] A. Nitaj, D/'{e}termination de courbes elliptiques pour la conjecture`); l(` de Szpiro, Acta Arith., to appear.`); l(`[6] H.E. Rose, On some classes of elliptic curves with rank two or three,`); l(` Univ. of Bristol Math. research report no. PM-97-01, 1997.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(`[7] T. Womack, http://tom.womack.net/maths/torstab.htm`); l(` shaf denotes the (conjectured) order of the Shafarevich-Tate group.`); l(` In Aug. '97, the largest shaf I knew was 224^2=50176, an example sent`); l(`to me by deWeger (see [4]). This also gave the largest known ratio`); l(`shaf/sqrt(Nc)=6.983, where Nc=conductor=51636585. The cat. name is`); l(`cur=A51636585 and the cat. number (in the apecs starter cat.) is ncur=549\ .`); l(`The motivation for considering this ratio, now called the deWeger ratio,`); l(`is explained in [4].`); l(` However this example is eclipsed by one sent to me by Nitaj in`); l(`Jan. '98 (see [5]): shaf=1832^2 with deWeger ratio 42.265. The example is`); l(`cur=A6305720190 and (in the starter catalog for ver.>=4.36) ncur=577.`); l(` Here are examples of shaf=n^2 for n=2..8 (all with rank r4=0) taken`); l(`from [3] except for n=6 and 8. Cremona went systematically through his`); l(`cat. of curves with Nc<1000, and so these examples, except for n=6,8,`); l(`are the earliest. I found the examples for n=6,8 as follows. (Cremona`); l(`subsequently extended his search to 1000>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(`The Birch, Swinnerton-Dyer conj. says that`); l(``); l(` shaf = ey(r4,dd)*NN^2/(r4!*omeg*reg*cP)`); l(``); l(`The procedure ey(r4,dd) calculates the r4-th derivative of the L-function`); l(`of E at s=1 to dd digits, and of course in practice this eqn. is only`); l(`approximate. (In fact the procedure sha() loops on increasing dd until`); l(`the right side is reasonably close to the square of an integer.) This`); l(`suggests that large NN should give large shaf, "other things being equal".`); l(`This is borne out in practice to some extent, except that the large shaf`); l(`is usually that of an isogenous curve with smaller NN --- so much for`); l(`the heuristics. exam(12,1,-2);isog();... leads to the shaf=36 example;`); l(`exam(8,-1/3);isog();... leads to shaf=64.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` shaf | cur | ncur | shaf/sqrt(Nc) | Cremona's cat. no.`); l(`======================================================================`); l(` 4 | H66 | 186 | .4924 | 66B3`); l(` 9 | C182 | 554 | .6671 | 182B3`); l(` 16 | G210 & H210 | 489 & 490| 1.104 | 210E8 & E7`); l(` 25 | C275 | 557 | 1.508 | 275B3`); l(` 36 | M2310 | 565 | .7490 | 2310T7`); l(` 49 | B546 | 559 | 2.097 | 546F2`); l(` 64 | F & G 1230 | 574 & 575| 1.825 | 1230F7 & F8`); l(``); l(`All the E in [4] have, apparently, rank r4=0. To find early examples of`); l(`shaf>1 and r4>0 I first tried the curves x^3+y^3=m as in [2],`); l(`using Gcub(1,0,0,1,0,0,0,0,0,-m);Isog(),.., but Nc soon becomes too big`); l(`to be managed by apecs. `); l(`So I turned to the family of curves in [1], namely`); l(` y^2+xy=x^3-16m*x^2-8m*x-m.`); l(`and of course curves isog. to these. For m=10 (resp. m=15*13) we get`); l(`Nc=1610, shaf=4 (resp. Nc=608595, shaf=16). Both have rank r4=1.`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` In [6], Rose has much interesting data on shaf for the following two`); l(`families: C_p:y^2=x^3+px and D_p:y^2=x^3-2px where p is a prime ==1 mod 8.`); l(`For example, C_51137 has r4=2 and shaf=9. His table S7 lists examples of`); l(`D_p with r4=1 and shaf>1. Of course these curves have large Nc: for C_p,`); l(`Nc=64p^2 and for D_p, Nc=256p^2.`); l(` Womack [7] lists examples of E with smallest known Nc with specified`); l(`torsion and shaf in the ranges 4<=NN<=16, 1<=shaf<=25; all have r4=0.`); l(`[7] was also an important source of examples for Exam(4).`); else l(` The intention is to have a handy compendium of examples that set`); l(`records or are remarkable in some way. Any e-mailed to me would be much`); l(`appreciated.`); l(` Nitaj (see reference [5] in Exam(5);) has obtained many interesting`); l(`E with high Szpiro ratio log(abs(DD))/log(Nc) (calc. in apecs by Szp();).`); l(`His best example to date, and one believes the highest Szpiro ratio known,`); l(`is A2526810 with ncur=581 (in the starter catalog for apecs ver.>= 4.36)`); l(`with Szpiro ratio = 8.81194.`); fi; else iee:=1:exam(args); fi; iee:=0:NULL; end: exam:=proc() local b,c,cuu,d,e,f,i,l,t; l:=proc();lprint(args);end; cuu:=`This value of the parameter is not allowed (makes DD=infinity).`; if nargs=0 then i:=0; l(` args || topic`); l(` -------||--------------------------------------`); l(` 1 || possible torsion`); l(` 2 || complex multiplication`); l(` 3 || isogenies`); l(` 4 || large rank`); l(` 5 || Shafarevich-Tate groups`); l(` * 6 || misc. (e.g.height, regulator, Szpiro ratio)`); l(` n,xx || where xx is one or more args, generates`); l(` || examples with torsion subgroup of order n`); l(` || (at least). See Exam(1) for details.`); l(` * barely begun`); else i:=args[1]: fi; if nargs=1 then if not type(i,posint) or i>6 then print(qts):RETURN();fi; if i=1 then print(Y^2+(1-c)*X*Y-b*Y=X^3-b*X^2); l(`with the parameters b,c chosen as follows:`); l(` T || b || c || example`); l(`----||-----------------------||------------------||----------------`); l(` C4|| b || 0 ||b=-1: A15`); l(` C5|| b || b ||b=1: A11`); l(` C6|| c+c^2 || c ||c=-1/2: A14`); l(` C7|| d^3 - d^2 || d^2 - d ||d=-1: D26`); l(` C8|| (d - 1)(2d - 1) || (d-1)(2d-1)/d ||d=1/6: F15`); l(` C9||e^5 - 2e^4 + 2e^3 - e^2|| e^3 - e^2 ||e=-1: B54`); l(` C10||e^3(1 - 3e + 2e^2)/ ||-e(1-3e+2e^2)/ ||e=1/3:I66`); l(` || (1 - 3e + e^2)^2 || (1-3e+e^2) ||`); l(` C12||f^(-4)(1 - f)(1 - 2f)* ||f^(-3)(1-f)(1-2f)*||f=1/3:G90`); l(` || (1-2f+2f^2)(1-3f+3f^2)|| (1-3f+3f^2)||`); l(` `); l(`Command to generate examples: Exam(n,x) where 4<=n<=10 or n=12 and`); l(`x is the value of the corresponding param. b,b,c,...,f.`); l(`The examples in this table and the following have the smallest possi\ ble`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(`conductor for the given T; however in some cases a simple transforma\ tion`); l(`is needed to put the equation in standard minimal Weierstrass form \ ---`); l(`after Ein(1-c,-b,-b,0,0) with appropriate b,c, Trwc() displays the t\ ransf.`); l(` `); l(` For C2XCn, n=4,6,8, choose the parameter for Cn as follows:`); l(` T || parameter || example`); l(`-------||-------------------------||-------------------`); l(` C2XC4 || b = (t^2 - 1)/16 || t = 3/5: B15`); l(` C2XC6 || c = (2t - 10)/(9 - t^2) || t = -1: B30`); l(` C2XC8 || d = (2 + t)/(2 - t^2) || t = 1: A210`); l(` `); l(`Command to generate examples: Exam(n,1,t) where n=8,12 or 16, the 1`); l(`is a dummy arg. to indicate a non-cyclic case, and t is the param.`); elif i=2 then l(` CM jay || E || E' || d || D `); l(`----------------------||-------||-------||-----||--------`); l(` 12^3=1728 || A32 || B32 || 2 || -4`); l(` 20^3=8000 || A256 || B256 || 2 || -8`); l(` 0 || A27 || B27 || 3 || -3`); l(` -15^3 || A49 || C49 || -7 || -7`); l(` -32^3 || D121 || E121 || -11 || -11`); l(` -(32*3)^3 || A361 || B361 || -19 || -19`); l(` -(64*3*5)^3 || A1849 || B1849 || -43 || -43`); l(` -(32*3*5*11)^3 || A4489 || B4489 || -67 || -67`); l(` -(64*3*5*23*29)^3 || A26569|| B26569||-163 || -163`); l(` 66^3 || C32 || D32 || -4 || -16`); l(` 2*30^3 || B36 || D36 || 3 || -12`); l(` (3*5*17)^3 || B49 || D49 || -7 || -28`); l(` -3*(32*5)^3 || C27 || D27 || -27 || -27`); l(``); l(` The list of jay's is stored in CMj and the corresponding d's in CMd.`); elif i=3 then l(` *--N--* || N=2,3,5,7,11,13,17,19,37,43,67,163`); l(` *--M--*--N--* || M=N=3,5`); l(`*--L--*--M--*--N--*|| L=M=N=3`); l(` *--M--* ||`); l(` | | ||`); l(` N N || (M,N)=(2,3),(2,5),(2,7),(3,5),(3,7)`); l(` | | ||`); l(` *--M--* ||`); l(` *--2--*--2--* *--3--*--3--* *--2--*--2--*`); l(` | | | | |`); l(` 2 2 2 2 2`); l(` | | | | |`); l(` * *--3--*--3--* *--2--*--2--*`); l(` * * *----3----*`); l(` | | | |`); l(` 2 2 2 2`); l(` | | | |`); l(` *--2--*--2--*--2--*--2--* *----3----*`); l(` | /| /|`); l(` 2 / 2 2 2`); l(` | 2 | / |`); l(` * / *---/3----*`); l(` *----3----*`); elif i=4 then l(`ncur|| args || conductor Nc || T || r`); l(`----||--------||---------------------------------||-----||-----`); l(` 77|| -- || 37 || O || 1`); l(` 177|| -- || 65=5*13 || C2 || 1`); l(` 270|| 3,-4,1 || 91=7*13 || C3 || 1`); l(` 370|| 4,-1/3 || 117=3^2*13 || C4 || 1`); l(` 371||4,15/2,6|| 117=3^2*13 ||C2xC2|| 1`); l(` 399|| 5,-3 || 123=3*41 || C5 || 1`); l(` 492|| 6,1/4 || 130=2*5*13 || C6 || 1`); l(` 502|| 7,8 || 574=2*7*41 || C7 || 1`); l(` 496||8,1,4/3 || 336=2^4*3*7 ||C2xC4|| 1`); l(` 504|| 8,7/6 || 966=2*3*7*23 || C8 || 1`); l(` 585|| -- || 389 || O || 2`); l(` 510|| -- || 1088=2^6*17 || C2 || 2`); l(` 449|| -- || 5077 || O || 3`); l(`...............................................................`); l(` 512|| 9,-1/3 || 1482=2*3*13*19 || C9 || 1`); l(` 517|| 10,-2 || 6270=2*3*5*11*19 || C10 || 1`); l(` continued ...`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` 529||12,1,-1/7|| 2310=2*3*5*7*11 ||C2xC6|| 1`); l(` 521|| 12,-1/2 || 4290=2*3*5*11*13 || C12 || 1`); l(` 540|| 16,1,3 || 82110=2*3*5*7*17*23 ||C2xC8|| 1`); l(` 515|| 3,-7,7 || 1862=2*7^2*19 || C3 || 2`); l(` 597|| 4,-1/45 || 2175=3*5^2*29 || C4 || 2`); l(` 609|| -- || 3264=2^6*3*17 ||C2xC2|| 2`); l(` 599|| 5,22 || 5302=2*11*241 || C5 || 2`); l(` 613|| 6,1/28 || 15022=2*7*29*37 || C6 || 2`); l(` 615|| 7,5/13 || 513110=2*5*13*3947 || C7 || 2`); l(` 601||8,1,19/7 || 41496=2^3*387813*19 ||C2xC4|| 2`); l(` 616|| 8,23/22 || 253506=2*3*11*23*167 || C8 || 2`); l(` 618|| 9,11/4 || 14049882=2*3^3*7*11*31*109 || C9 || 2`); l(` 619|| 10,1/12 || 1474770=2*3*5*11*41*109 || C10 || 2`); l(` 604||12,1,7/6 || 356730=2*3*5*11*23*47 ||C2xC6|| 2`); l(` 620|| 12,5/11 || 36817770=2*3*5*11*31*59*61 || C12 || 2`); l(` 603||16,1,3/5 || 169350090=2*3*5*7*13*17*41*89 ||C2xC8|| 2`); l(` 608|| -- || 80256=2^7*3*11*19 || C2 || 3`); l(` 596||3,-31,31 || 474734=2*13*19*31^2 || C3 || 3`); l(` 610|| -- || 562224=2^4*3*13*17*53 ||C2xC2|| 3`); l(`>>>>>HIT SEMICOLON KEY ; FOR NEXT SCREEN`);readstat(); l(` 611|| 4,29/45 || 1107075=3*5^2*29*509 || C4 || 3`); l(` 612|| 5,49/43 || 1362025=5^2*7*43*181 || C5 || 3`); l(` 614|| 6,85/4 || 11634970=2*5*17*89*769 || C6 || 3`); l(` 627||7,-19/21 || 399562590=2*3*5*7*19*239*419 || C7 || 3`); l(` 617||8,1,13/51|| 1092624=2^4*3*13*17*103 ||C2xC4|| 3`); l(` 628||8,-10/21 || 779644110=2*3*5*7*23*31*41*127 || C8 || 3`); l(` 629||12,1,6/29|| 2*3^2*5*7*11*13*17*23*29*31*139||C2xC6|| 3`); l(` 588||16,1,5/12|| 2*3*5*7*17*29*79*263*433 ||C2xC8|| 3`); l(` 537|| -- || 234446=2*117223 || O || 4`); l(` 625|| -- || 7890368=2^6*31*41*97 || C2 || 4`); l(` 624||3,-109,421 274996358=2*7*13*37*97*421 || C3 || 4`); l(` 551|| -- || 19047851=prime || O || 5`); l(` 626|| -- || 4378153024=2^6*7*47*337*617 || C2 || 5`); l(` 605|| -- || 5258110041=3^2*584234449 || O || 6`); l(` 606|| -- || 1074680679376=2^4*199*337525339|| O || 7`); l(` 621|| -- || ~5.617*10^14 || O || 8`); l(` 622|| -- || ~4.84*10^17 || O || 9`); l(` 623|| -- || ~1.97*10^21 || O || 10`); l(` 457|| -- || ~1.80*10^24 || O || 11`); l(` 458|| __ || ~2.7*10^29 || O || 12`); l(` 459|| __ || ~3.66*10^38 || O || 13`); l(` 460|| __ || ~3.6*10^37 || O || 14`); else `not programmed`; fi; elif i=2 then l(`Examples with T=C2 not explicitly programmed. Any E with`); l(`a3=a6=0 is an example with T=C2 (at least).`); elif nargs=2 then t:=args[2]: if i=4 then examQ(t,0); elif i=5 then examQ(t,t); elif i=6 then examQ(t+t^2,t); elif i=7 then examQ(t^3-t^2,t^2-t); elif i=8 then if t=0 then eprint(cuu); else examQ((t-1)*(2*t-1),(t-1)*(2*t-1)/t); fi; elif i=9 then examQ(t^5-2*t^4+2*t^3-t^2,t^3-t^2); elif i=10 then if 1-3*t+t^2=0 then eprint(cuu); else examQ(t^3*(1-3*t+2*t^2)/(1-3*t+t^2)^2,-t*(1-3*t+2*t^2)/(1-3*t+t^2)); fi; elif i=12 then if t=0 then eprint(cuu); else examQ((1-t)*(1-2*t)*(1-2*t+2*t^2)*(1-3*t+3*t^2)/t^4,\ (1-t)*(1-2*t)*(1-3*t+3*t^2)/t^3); fi; else print(qts):RETURN(); fi; elif nargs=3 then b:=args[2]:c:=args[3]: if i=3 then if c=0 or 27*c=b^3 then fprint(`These values of the parameters give a singular curve:`); fi; examQ(b,0,c,0,0): elif i=4 then if c=0 or 4*c^2=b^2 then fprint(`These values of the parameters give a singular curve:`); fi; examQ(0,b,0,b^2/4-c^2,0); elif b=1 then if i=8 then examQ((c^2-1)/16,0); elif i=12 then if c^2=9 then eprint(cuu); else d:=(2*c-10)/(9-c^2):examQ(d+d^2,d); fi; elif i=16 then if c=-2 or c^2=2 then eprint(cuu); else d:=(2+c)/(2-c^2):examQ((d-1)*(2*d-1),(d-1)*(2*d-1)/d); fi; else print(qts):RETURN(); fi; else print(qts):RETURN(); fi; elif nargs>0 then print(qts):RETURN(); fi; NULL; end: examQ:=proc() local a,e,j;global iee,K_,U,V; e:=iee:iee:=0:j:=sex_([args]); if nops(j)=2 then a:=1-j[2],-j[1],-j[1],0,0; else a:=op(j): fi; if type(j,list(rational)) then K_:=1; elif type(j,list(ratpoly(rational,T))) then K_:=T; elif type(m_,integer) and type(j,list(polynom(rational,w_))) then K_:=m_; else K_:=0; fi; if K_=1 then ein(a); else ell(a,K_); fi; if e=1 then iee:=e; if sT=4 then l(`This value of the parameter gives a singular curve:`); elif bouu then print(cus,cur); fi; We(); fi; if K_=1 and DD<>0 and NN<1 and nops(bic)=1 then U:=0;V:=0;j:=bic[1]:j:=[j[5]/j[6],j[7]/j[8]]:U:='U';V:='V'; if e=1 then Tor(j);else tor(j);fi; elif tor_able then if e=1 then print(`We determine the torsion subgroup:`);Tor(); else tor(); fi; fi; end: