This is TeX, Version 3.14159 (Web2C 7.3) (format=latex 1999.4.5) 8 AUG 1999 17:07 **joe_jjprime.tex (joe_jjprime.tex LaTeX2e <1998/12/01> Babel and hyphenation patterns for american, french, german, ngerman, n ohyphenation, loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 1999/01/07 v1.4a Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size10.clo File: size10.clo 1999/01/07 v1.4a Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (joe_jjprime.aux) \openout1 = `joe_jjprime.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 2. LaTeX Font Info: ... okay on input line 2. Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 I worked out the X_0(11) information for you. The short ans wer is that I[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 can parameterize X_0(11) in terms of j and a variable I call x by:[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 [] \OT1/cmtt/m/n/10 + (-22*j - 431860)*x^6 + (297*j - 123816)*x^5 + (1309*j + 2299407)*x^4[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 [] \OT1/cmtt/m/n/10 + (-j^2 + 5732*j + 3274332)*x + (16*j^2 + 6969*j + 912673) = 0[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 [] \OT1/cmtt/m/n/10 j' = (-754*x^15 - 23752655*x^14 - 29613745966*x^13 - 62440 36882207*x^12 -[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 376104349304354*x^11 + (-j - 7114347826251259)*x^10 + (-1967 07*j -[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 24131393248600286)*x^9 + (-873240811*j + 264386461447029845) *x^8 +[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-267091946884*j + 1012641259503839482)*x^7 + (-172182226651 69*j -[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 2587403021099187533)*x^6 + (-283597926046359*j - 10441866585 623711194)*x^5 +[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-579172383588254*j + 2074752923950835587)*x^4 + (5999011646 737623*j +[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 35578660053077953178)*x^3 + (10293966016026733*j + 206804447 32221011671)*x^2[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 + (-16583607774597939*j - 13671051080851018906)*x + (-263952 08710078913*j -[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 6903889144381374585))/(-x^15 + 137*x^14 - 7143*x^13 + 114009 *x^12 +[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 5734630*x^11 - 406417441*x^10 + 12628650656*x^9 - 2461244490 24*x^8 +[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 634871630266368*x^4 + 1066824779169792*x^3 + 289390601437184 *x^2 -[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 The rest of this email is a log of my work. It contains a l ot more[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 information, including a different parameterization for X_0( 11) in terms of[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 usual functions x,y on the elliptic curve. I leave it to yo u to work out[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 ------------------------ LOG OF PARI SESSION -------------- ------[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ From Cremona's tables we find that X_0(11) is the ellipti c curve[] [] [1 ] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ Now, I don't know how to directly get the j and j' functi ons on[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ X_0(11), so we start by expressing everything in terms of q-expansions.[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ The easiest thing is to find the q-expansion for the cano nical differential[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %3 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4 , -2, 4, 0, 2,[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 2, -2, -1, 0, -4, -8, 5, -4, 0, 2, 7, 8, -1, 4, -2, -4, 3, 0 , -4, 0, -8,[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 -4, -6, 2, -2, 2, 8, 4, -3, 8, 2, 8, -6, -10, 1, 0, 0, 0, 5, -2, 12, -14,[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 4, -8, 4, 2, -7, -4, 1, 4, -3, 0, 4, -6, 4, 0, -2, 8, -10, - 4, 1, 16, -6,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 4, -2, 12, 0, 0, 15, 4, -8, -2, -7, -16, 0, -8, -7, 6, -2, - 8, 2, -4, -16,[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0, 2, 12, 18, 10, 10, -2, -3, 8, 9, 0, -1, 0, -8, -10, 4, 0, 1, -24, 8, 14,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 -9, -8, 8, 0, 6, -8, -18, -2, 0, 14, 5, 0, -7, -2, 10, -4, - 8, 6, 4, 8, 0,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 -8, 3, 6, -10, -8, 2, 0, 4, 4, 7, -8, -7, 20, 6, 8, 2, -2, 4 , -16, -1, 12,[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 -12, 0, 3, 4, 0, -12, -6, 0, 8, -4, -5, -30, -15, -4, 7, 16, -12, 0, 3, 14,[] [] [2] Overfull \hbox (17.24684pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ OK, first we double-check against the coefficients we did by hand:[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 [[1, 2, 4, 5, 8, 1, 7, -11], [-1, -3, -7, -13, -17, -26, -19 , -37]][] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 [[1, 2, 4, 5, 8, 1, 7, -11, 10], [-1, -3, -7, -13, -17, -26, -19, -37, 15]][] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 [[1, 2, 4, 5, 8, 1, 7, -11, 10, -12], [-1, -3, -7, -13, -17, -26, -19, -37,[] [] [3] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 20245856256*q^4 + 333202640600*q^5 + 4252023300096*q^6 + 446 56994071935*q^7[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 + 401490886656000*q^8 + 3176440229784420*q^9 + 2256739330959 3600*q^10 +[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ in terms of x and y. Let's do j' first. Note that j' is a degree 12[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ function on X_0(11). So we can write j' = (a(x)+y*b(x))/ c(x) where[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ But from the q-expansion of j' we know it has a pole of o rder 11 at[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ (This why we do j' first. After we do j' we will know th e other cusp[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ let's make a place to store the coefficients for coeffici ent matching.[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 ? for(n=0,9, xnq=yq*xq^n; for(k=1,150, m[k,n+1+12+1]=coeff( xnq,k-13)))[] [] [4] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 [-3308 -1781 6780 2704 -643 -148 11 0 0 0 0 0 0 353 2170 103 1 -697 23 1 0 0[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0][] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ so, we have found the coefficients of the expression for j' in terms[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %59 = 11*x^6 - 148*x^5 - 643*x^4 + 2704*x^3 + 6780*x^2 - 178 1*x - 3308[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ j' = (11*x^6 - 148*x^5 - 643*x^4 + 2704*x^3 + 6780*x^2 - 1781*x - 3308[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ + y*(x^5 + 23*x^4 - 697*x^3 + 1031*x^2 + 2170*x + 353)) / (-x + 16)[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ By the way, we really should have been able to predict th at ajj and bjj[] [] Overfull \hbox (59.24648pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ would have degree 6 and 5, in the following way: jj has a pole of order 11[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ at \infty, while x and y have poles of order 2 and 3. Si nce we know the[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ denominator is linear (ie: pole of order 2) the numerator has to have a[] [] [5] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ So we see that jjxy(16,-61) is infinity, so this is the o ther cusp.[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %72 = q^-11 + 744 + 196884*q^11 + 21493760*q^22 + 864299970* q^33 +[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 44656994071935*q^77 + 401490886656000*q^88 + 317644022978442 0*q^99 +[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ Since j has a pole of order 1 at \infty and the denominat or has a pole[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ of order 22, the numerator must have a pole of order 23, so deg(aj)<=11,[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 ? for(n=0,10, xnq=yq*xq^n; for(k=1,150, m[k,n+1+11+1]=coeff( xnq,k-23)))[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 6419026712245052, -15124441025165095, -2179206296697095, 558 818644494837,[] [] [6] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 5999011646737623, -579172383588254, -283597926046359, -17218 222665169,[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 6419026712245052, -15124441025165095, -2179206296697095, 558 818644494837,[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 5999011646737623, -579172383588254, -283597926046359, -17218 222665169,[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %95 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %96 = 743*x^11 + 21559873*x^10 + 19161808636*x^9 + 253587651 7772*x^8 +[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 15124441025165095*x^4 + 6419026712245052*x^3 + 8488729414584 3033*x^2 +[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 5999011646737623*x^3 + 10293966016026733*x^2 - 1658360777459 7939*x -[] [] Overfull \hbox (53.99652pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ suppose you want to find the j-invariants 11-isogenous to jj=12 mod 1117.[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %100 = (11*x^6 + (y - 148)*x^5 + (23*y - 643)*x^4 + (-697*y + 2704)*x^3 +[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (1031*y + 6780)*x^2 + (2170*y - 1769)*x + (353*y - 3500))/(- x + 16)[] [] [7] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %101 = 11*x^6 + (y - 148)*x^5 + (23*y - 643)*x^4 + (-697*y + 2704)*x^3 +[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %104 = (xx^5 + 23*xx^4 - 697*xx^3 + 1031*xx^2 + 2170*xx + 35 3)*y + (11*xx^6[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %149 = -x^13 + 76*x^12 - 2346*x^11 + 37268*x^10 - 313775*x^9 + 1216776*x^8[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 - 526724*x^7 - 6793732*x^6 - 4239147*x^5 + 31933000*x^4 + 76 226018*x^3 +[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %150 = x^12 - 60*x^11 + 1386*x^10 - 15092*x^9 + 72303*x^8 - 59928*x^7 -[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 432124*x^6 - 120252*x^5 + 2315115*x^4 + 5108840*x^3 + 551542 2*x^2 +[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 [mod(1, 1117)*x^10 + mod(129, 1117)*x^9 + mod(789, 1117)*x^8 + mod(372,[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 1117)*x^7 + mod(341, 1117)*x^6 + mod(560, 1117)*x^5 + mod(62 4, 1117)*x^4 +[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 mod(429, 1117)*x^3 + mod(577, 1117)*x^2 + mod(205, 1117)*x + mod(788, 1117) 1][] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %160 = (11*x^6 + (y - 148)*x^5 + (23*y - 643)*x^4 + (-697*y + 2704)*x^3 +[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (1031*y + 6780)*x^2 + (2170*y + (jval - 1781))*x + (353*y + (-16*jval -[] [] [8] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %161 = 11*x^6 + (y - 148)*x^5 + (23*y - 643)*x^4 + (-697*y + 2704)*x^3 +[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (1031*y + 6780)*x^2 + (2170*y + (jval - 1781))*x + (353*y + (-16*jval - 3308))[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %162 = -x^13 + 76*x^12 - 2346*x^11 + 37268*x^10 - 313775*x^9 + 1216776*x^8[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 + (22*jval - 526988)*x^7 + (-649*jval - 6785944)*x^6 + (3443 *jval -[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 4280463)*x^5 + (27049*jval + 31608412)*x^4 + (-85151*jval + 77247830)*x^3 +[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (jval^2 - 206196*jval + 87377988)*x^2 + (-32*jval^2 + 84743* jval +[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %163 = x^12 - 60*x^11 + 1386*x^10 - 15092*x^9 + 72303*x^8 - 59928*x^7 +[] [] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 2299407)*x^4 + (-6105*jval + 5182100)*x^3 + (-12529*jval + 5 665770)*x^2 +[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-jval^2 + 5732*jval + 3274332)*x + (16*jval^2 + 6969*jval + 912673)[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ So far this has worked very well. In fact this is an alt ernate equation[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %164 = (xx^5 + 23*xx^4 - 697*xx^3 + 1031*xx^2 + 2170*xx + 35 3)*x + (11*xx^6[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 - 148*xx^5 - 643*xx^4 + 2704*xx^3 + 6780*xx^2 + (jval - 1781 )*xx +[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %165 = (-11*x^6 + 148*x^5 + 643*x^4 - 2704*x^3 - 6780*x^2 + (-jval +[] [] Overfull \hbox (64.49643pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 1781)*x + (16*jval + 3308))/(x^5 + 23*x^4 - 697*x^3 + 1031*x ^2 + 2170*x + 353)[] [] Overfull \hbox (27.74675pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 ? yxjj(xval,yval)=(-11*xval^6 + 148*xval^5 + 643*xval^4 - 27 04*xval^3 -[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 6780*xval^2 + (-jval + 1781)*xval + (16*jval + 3308))/(xval^ 5 + 23*xval^4 -[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 %166 = (-754*x^15 - 23752655*x^14 - 29613745966*x^13 - 62440 36882207*x^12 -[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 376104349304354*x^11 + (-jval - 7114347826251259)*x^10 + (-1 96707*jval -[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 24131393248600286)*x^9 + (-873240811*jval + 2643864614470298 45)*x^8 +[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-267091946884*jval + 1012641259503839482)*x^7 + (-172182226 65169*jval -[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 6903889144381374585))/(-x^15 + 137*x^14 - 7143*x^13 + 114009 *x^12 +[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 5734630*x^11 - 406417441*x^10 + 12628650656*x^9 - 2461244490 24*x^8 +[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 634871630266368*x^4 + 1066824779169792*x^3 + 289390601437184 *x^2 -[] [] [9] Overfull \hbox (38.24666pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ So, there you go. The short answer is to plug the initia l value for j[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ x^12 - 60*x^11 + 1386*x^10 - 15092*x^9 + 72303*x^8 - 59928*x^7 +[] [] Overfull \hbox (22.4968pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-22*j - 431860)*x^6 + (297*j - 123816)*x^5 + (1309*j + 2299 407)*x^4 +[] [] Overfull \hbox (17.24684pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-6105*j + 5182100)*x^3 + (-12529*j + 5665770)*x^2 + (-j^2 + 5732*j +[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 \\ j' = (-754*x^15 - 23752655*x^14 - 29613745966*x^13 - 624 4036882207*x^12[] [] Overfull \hbox (11.99689pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 - 376104349304354*x^11 + (-j - 7114347826251259)*x^10 + (-19 6707*j -[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 24131393248600286)*x^9 + (-873240811*j + 264386461447029845) *x^8 +[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 (-267091946884*j + 1012641259503839482)*x^7 + (-172182226651 69*j -[] [] Overfull \hbox (43.49661pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 2587403021099187533)*x^6 + (-283597926046359*j - 10441866585 623711194)*x^5[] [] Overfull \hbox (32.9967pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 + (-579172383588254*j + 2074752923950835587)*x^4 + (59990116 46737623*j +[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 20680444732221011671)*x^2 + (-16583607774597939*j - 13671051 080851018906)*x[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 + (-26395208710078913*j - 6903889144381374585))/(-x^15 + 137 *x^14 -[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 7143*x^13 + 114009*x^12 + 5734630*x^11 - 406417441*x^10 + 12 628650656*x^9 -[] [] Overfull \hbox (1.49698pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 175076351672320*x^5 - 634871630266368*x^4 + 1066824779169792 *x^3 +[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 ____________________________________________________________ _______________[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 [] \OT1/cmtt/m/n/10 Joseph Loebach Wetherell jlwether@alum. mit.edu[] [] Overfull \hbox (6.74693pt too wide) in paragraph at lines 448--448 [] \OT1/cmtt/m/n/10 wetherel@math. usc.edu[] [] Overfull \hbox (48.74657pt too wide) in paragraph at lines 448--448 []\OT1/cmtt/m/n/10 ___________________=====================================____ _______________[] [] [10] (joe_jjprime.aux) ) Here is how much of TeX's memory you used: 196 strings out of 10898 1939 string characters out of 72048 58005 words of memory out of 263001 3199 multiletter control sequences out of 10000+0 3808 words of font info for 15 fonts, out of 200000 for 1000 15 hyphenation exceptions out of 1000 23i,4n,17p,135b,300s stack positions out of 300i,100n,500p,30000b,4000s Output written on joe_jjprime.dvi (10 pages, 22540 bytes).