## [2004.10.09]modular$B6J@~(BX_0(11)$B$N(BHeegner$BE@(B

### X0(11)$B$N(BHeegner$BE@$r7W;;$9$k%W%m%0%i%(B(pari/GP)

$B"#(Bgenus 1$B$N(Bmodular$B6J@~(BX0(11)$B$K(BQ-isomorphic$B$J(Bconductor 11$B$NBJ1_6J@~(B
E: y2+y = x3-x2-10x-20
$B$N(BHeegner$BE@$r5a$a$k!#(B

$B"#BJ1_6J@~(BE$B$N$M$8$lE@72$HM-M}E@72$r5a$a$F$*$/!#(B
pari/gp$B$H(Bmwrank3$B$K$h$j!"(BE$B$N$M$8$lE@72$H(Brank$B$O!"(B
E(Q)tors = Z/5Z,
rank E(Q) = 0
$B$G$"$k$3$H$,J,$+$k!#(B $B$h$C$F!"(BE$B$N(BMordel-Weil$B72$O!"(B E(Q) = Z/5Z = { O, [5,5], [16,-61], [16,60], [5,-6]} $B$G$"$k!#(B [pari/gp$B$K$h$k7W;;(B] gp> e=ellinit([0,-1,1,-10,-20]) time = 188 ms. %1 = [0, -1, 1, -10, -20, -4, -20, -79, -21, 496, 20008, -161051, -122023936/161051, [4.346308158205394421969490790, -1.673154079102697210984745395 + 1.320848922269075665602515977*I, -1.673154079102697210984745395 - 1.320848922269075665602515977*I]~, 1.269209304279553421688794616, 0.6346046521397767108443973083 + 1.458816616938495229330889613*I, -1.318701713826012913940217413 + 0.E-29*I, -0.6593508569130064569701087069 - 3.990938775365904887253619872*I, 1.851543623455959317708006712] gp> ellglobalred(e) time = 1 ms. %2 = [11, [1, 0, 0, 0], 5] gp> elltors(e,1) time = 34 ms. %3 = [5, [5], [[5, 5]]] gp> for(i=1,5,print(ellpow(e,[5,5],i))) [5, 5] [16, -61] [16, 60] [5, -6] [0] time = 11 ms.  [mwrank3$B$K$h$k7W;;(B] bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work,
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0,-1,1,-10,-20]

Curve [0,-1,1,-10,-20] :        No points of order 2
Basic pair: I=496, J=40016
disc=-1113184512
By Lemma 5.1(a), 2-adic index = 1
One (I,J) pair
Looking for quartics with I = 496, J = 40016
Looking for Type 3 quartics:
Trying positive a from 1 up to 3 (square a first...)
(1,0,-94,484,-695)      --trivial
(1,0,-28,44,-24)        --trivial
Trying positive a from 1 up to 3 (...then non-square a)
Trying negative a from -1 down to -9
(-2,-1,11,21,-13)       --trivial
(-2,2,-4,32,-28)        --trivial
(-6,2,2,14,-8)  --trivial
(-7,-12,2,16,1) --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 0
Selmer  rank contribution from B=im(eps) = 0
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 0
After descent, rank of points found is 0

The rank and full Mordell-Weil basis have been determined unconditionally.
Regulator = 1

(1.6 seconds)


$B"#5u#2Q(sqrt{-7})$B$NN?t$O(B1$B$G$"$k!#(B
K$B$N@0?t4D(BOK=Z[(-1+sqrt{-7})/2]$B$O!"(BM0(11)={((a b);(c d)) : a,b,c,d \in Z, 11|c }$B$N@0?t4D(BO=Z+Z((-4 -2); (11 5))$B$KF17?$G$"$k!#(B $B&S(B=(-9+sqrt{-7})/22$B$O!"(BO$B$NITF0E@(B(fixed point)$B$G$"$k!#      ((1 0);(0 1))$B&S(B=$B&S(B/1=$B&S(B, ((-4 -2);(11 5))$B&S(B=(-4$B&S(B-2)/(11$B&S(B+5)=(-9+sqrt{-7})/22=$B&S(B $B$H$J$k!#(B $B&S$OH=JL<0(B-7$B$N#22+9t+2=0$B$N:,$G$"$j!"(B 11$B&S$OH=JL<0(B-7$B$N#22+9s+22=0$B$N:,$G$"$k$N$G!"&S(B=(-9+sqrt{-7})22$B$O(BX0(11)$B$N(BHeegner$BE@$G$"$k!#(B modular$B7A<0(Bf$B$N(BFourier$B78?t(Ban(E)$B$O!"(BE mod p$B$NM-M}E@$+$i!"0J2<$NEy<0$h$j7W;;$G$-$k!#(B f=$B&2(Bn=1$B!g(B{anqn} = q$B&0(Bn=1$B!g(B{(1-q11n)2(1-qn)2} = q-2q2-q3+2q4+q5+2q6-2q7-2q9-2q10........ $B$3$3$G!"(Bq=e2$B&P(Bi$B&S(B$B$H$7$F!"J#AG?tBN(BC$B$NE@(B z = $B&2(Bn=11000{(an/n)qn}
$B$r(BWeierstrass uniformisation$B$G(BE(C)$B$KQ(sqrt{-7})-$BM-M}E@(BP((1-sqrt{-7})/2,-2-2sqrt{-7})$B$rF@$k!#(B

[pari/gp$B$K$h$k7W;;(B]
gp> read("x0_11.gp")
time = 67 ms.
gp>  tau=(-9+sqrt(-7))/22
time = 0 ms.
%1 = -9/22 + 0.12026142323020866320461889789269365571410269014011137183492429360004858287410380126183604029429743684613929901538624832253379681311941885383760866163091398739623412407856017692271316340295851185004445*I
gp>  q0=exp(2*Pi*I*tau)
time = 3 ms.
%2 = -0.39515111959617535063168342980195099288850717247298087491745329848478165253015833673093490270575974222138283855294957723309849798825224051360340125290343127651615544542994157742440587236904592845502550 - 0.25394820464883811117899444081730311657083474108403541216181985350498192575075778519563644745515743301261214180804701868781844515191378635355035959480821003763687835930828970804742530295453990240947869*I
gp>  f2(q0,1000)
time = 486 ms.
%3 = -0.50768372171182136867551784670181892208779689673224346718685476849216333544511108907614492236860504292465811851132851497491268013139073759059708560422829969941684297923544538412038379127552892113856282 - 0.40562904451604516529142753598929424354673787306147774542102120458224309134668996496697769442177074243769344356157311724198746365650385503909741351082777062649049008879082011123267657101004751995716840*I
gp>  p=ellztopoint(e,%3)
time = 59 ms.
%4 = [0.49999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 - 1.3228756555322952952508078768196302128551295915412250901841672296005344116151418138801964432372718053075322891692487315478717649443136073922136952779400538613585753648641619461498447974325436303504890*I, -1.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 - 5.2915026221291811810032315072785208514205183661649003607366689184021376464605672555207857729490872212301291566769949261914870597772544295688547811117602154454343014594566477845993791897301745214019561*I]
gp>  contfrac(real(p[1]))
time = 4 ms.
%5 = [0, 2]
gp> contfrac(imag(p[1])/sqrt(7))
time = 1 ms.
%6 = [-1, 2]
gp>  contfrac(real(p[2]))
time = 0 ms.
%7 = [-2]
gp>  contfrac(imag(p[2])/sqrt(7))
time = 0 ms.
%8 = [-2]


Gal(Q(sqrt{-7})/Q)$B$N<+L@$G$J$$85&R(B:sqrt{-7}B"*(B-sqrt{-7}BKBP7F!"(B PB&R(B((1+sqrt{-7})/2,-2+2sqrt{-7})Bb(BEBN(BQ(sqrt{-7})-BM-M}E@G"j!"(BR=P+PB&R(BBO!"&RN:nMQGITJQJNG!"(BEBN(BQ-BM-M}E@G"k!#(B BQ-BM-M}E@(BRBr7W;;9kH!"(B(16,-61)BHJk!#(B [pari/gpBKhk7W;;(B] gp> r7=Mod(w,w^2+7) time = 0 ms. %4 = Mod(w, w^2 + 7) gp> ellisoncurve(e,[(1-r7)/2,-2-2*r7]) time = 1 ms. %5 = 1 gp> elladd(e,[(1-r7)/2,-2-2*r7],[(1+r7)/2,-2+2*r7]) t6me = 4 ms. %6 = [Mod(16, w^2 + 7), Mod(-61, w^2 + 7)]  B^?!"(BP,2P((-3+sqrt{-7})/2,(-3+sqrt{-7})/2),3P((5-3sqrt{-7})/2,(3+9sqrt{-7})/2),4P((-11-17sqrt{-7})/2,-104-17sqrt{-17})BO(BKB@0E@G"k,!"(B5P(-3/4,(-4-11sqrt{-7})/8)BO(BQ(sqrt{-7})-B@0E@GOJ$$!#$h$C$F!"(BP$B$O0L?tL58B$N(BQ(sqrt{-7})-$BM-M}E@$G$"$k!#(B

[pari/gp$B$K$h$k7W;;(B]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],2)
time = 6 ms.
%19 = [Mod(1/2*w - 3/2, w^2 + 7), Mod(1/2*w - 3/2, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],3)
time = 12 ms.
%20 = [Mod(-3/2*w + 5/2, w^2 + 7), Mod(9/2*w + 3/2, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],4)
time = 4 ms.
%21 = [Mod(-17/2*w - 11/2, w^2 + 7), Mod(-17*w - 104, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],5)
time = 4 ms.
%22 = [Mod(-3/4, w^2 + 7), Mod(-11/8*w - 1/2, w^2 + 7)]
gp>  ellpow(e,[(1-r7)/2,-2-2*r7],6)
time = 3 ms.
%23 = [Mod(2319/5618*w - 1331/5618, w^2 + 7), Mod(234219/148877*w - 280499/148877, w^2 + 7)]


$B"#5u#2Q(sqrt{-6})$B$NN?t$O(B2$B$G$"$k!#(B
K$B$N@0?t4D(BZ[sqrt{-6}]$B$O!"(BM0(11)$B$N@0?t4D(BO=Z+Z((-4 -2);(11 4))$B$KF17?$G$"$k!#(B $B&S(B=(-4+sqrt{-6})/11$B$O!"(BO$B$NITF0E@$G$"$k!#(B $B&S$OH=JL<0(B-24$B$N#22+8t+2=0$B$N:,$G$"$j!"(B11$B&S$bH=JL<0(B-24$B$N#22+8s+22=0$B$N:,$G$"$k$N$G!"&S(B=(-4+sqrt{-6})/11$B$O(BX0(11)$B$N(BHeegner$BE@$G$"$k!#(B $B$3$3$G!"(Bq=e2$B&P(Bi$B&S(B$B$H$7$F!"J#AG?tBN(BC$B$NE@(B z = $B&2(Bn=11000{anqn/n}
$B$r(BWeierstrass uniformisation$B$G(BE(C)$B$KQ(sqrt{-6},sqrt{2})-$BM-M}E@(B P((-2-5sqrt{2})/2+(-3-4sqrt{2})sqrt{-6}/2,(-40-29sqrt{2})/2+(-7+sqrt{2})sqrt{-6}/2) $B$rF@$k!#(B

[pari/gp$B$K$h$k7W;;(B]
gp>  tau=(-4+sqrt(-6))/11
time = 0 ms.
%25 = -4/11 + 0.22268088570756164529066218860962649017872249824151546622115386975008730704157409332180540300951274861987223647514674446589626049634888703105688945011247986843899206795706150647705728229735561547486193*I
gp>  z0=f2(exp(2*Pi*I*tau),1000)
time = 381 ms.
%26 = -0.15882879211510579636430227271912781529638240464384567578493037113085848483539422706992217824565132107569758327069092235806887319449934046639144707260405449931839582904124657019634631138115048014466894 - 0.24453812691369989217080875059881915143692338996456510360871230053174983929209723562732673268059423630966634772698079666697599660082422064430139785789787914202528449809320916898198835407804721853282096*I
gp>  p=ellztopoint(e,z0)
time = 49 ms.
%27 = [-4.5355339059327376220042218105242451964241796884423701829416993449768311961552675971259688358191039318375346155772807425623120901396268430316103037427498395785330566648187639818894998762528819551514286 - 10.602437844450276321405711478082326555720142236226527704872266768684172633821172688134373896685954654930396467501548095807168078569752445927685297975233412423355742048900737185143344438310706355028282*I, -40.506096654409878207624486501040622139260242192965747061061856200865620937700552063330619247750802804657700770348228306861410122809835689583339761707949069555491728655948831094959099282266715339878286 - 6.8411632921722460501630479199647475049380109284879648214586170059264283041918025558083618291089922461894549628777429589070360840144331982168785883172361012863731013844445624172987305483471065208589195*I]
gp>  contfrac(real(p[1]))
time = 1 ms.
%28 = [-5, 2, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6, 1, 1, 6]
gp>  contfrac(imag(p[1])/sqrt(6))
time = 1 ms.
%29 = [-5, 1, 2, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22, 3, 5, 3, 22]
gp>  contfrac(real(p[2]))
time = 1 ms.
%30 = [-41, 2, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 1, 1, 40, 4]
gp>  contfrac(imag(p[2])/sqrt(6))
time = 2 ms.
%31 = [-3, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1, 1, 4]


$B=[4DO"J,?t(B[6$B!&(B,1,1$B!&(B]=$B&A$O!"#2 $B&A(B = 6+1/(1+1/(1+1/$B&A(B)) $B$D$^$j!"(B 2$B&A(B2-12$B&A(B-7 = 0 $B$N@5:,$J$N$G!"(B
$B&A(B = (6+5sqrt{2})/2 $B$rF@$k!#$h$C$F!"=[4DO"J,?t(B[-5,2,6$B!&(B,1,1$B!&(B]=[-5,2,$B&A(B]$B$r5a$a$k$H!"(B [-5,2,$B&A(B] = -5+1/(2+1/$B&A(B) = (-2-5sqrt{2})/2 $B$H$J$k!#(B $B$^$?!"(B[22$B!&(B,3,5,3$B!&(B]=$B&B$O!"#2      $B&B(B = 22+1/(3+1/(5+1/(3+1/$B&B(B))) = (1138$B&B(B+357)/(51$B&B(B+16)
$B$D$^$j!"(B
$B&B(B2-22$B&B(B-7 = 0
$B$N@5:,$J$N$G!"(B $B&B(B = 11+8sqrt{2}
$B$rF@$k!#$h$C$F!"(B
[-5,1,2,22$B!&(B,3,5,3$B!&(B] = [-5,1,2,$B&B(B] = (-13$B&B(B-4)/(3$B&B(B+1) = (-3-4sqrt{2})/2 $B$H$J$k!#(B $BF1MM$K$7$F!"(B [40$B!&(B,1,1$B!&(B] = (40+29sqrt{2})/2 [-40,2,40$B!&(B,1,1$B!&(B] = (-40-29sqrt{2})/2 [4$B!&(B,1$B!&(B] = 2+2sqrt{2} [-3,4$B!&(B,1$B!&(B] = (-7-sqrt{2})/2 $B$rF@$k!#(B

[pari/gp$B$K$h$k7W;;(B]
gp>  r2=Mod(w,w^2-2)
time = 1 ms.
%1 = Mod(w, w^2 - 2)
gp>  -5+1/(2+1/((6+5*r2)/2))
time = 17 ms.
%2 = Mod(-5/2*w - 1, w^2 - 2)
gp>  contfracpnqn([22,3,5,3,b])
time = 0 ms.
%3 =
[1138*b + 357 1138]

[51*b + 16 51]

gp>  b-(1138*b + 357)/(51*b + 16)
time = 1 ms.
%4 = (51*b^2 - 1122*b - 357)/(51*b + 16)
gp>  factor(51*b^2 - 1122*b - 357)
time = 6 ms.
%5 =
[b^2 - 22*b - 7 1]

gp>  contfracpnqn([-5,1,2,b])
time = 0 ms.
%6 =
[-13*b - 4 -13]

[3*b + 1 3]

gp>  subst((-13*b-4)/(3*b+1),b,(11+8*r2))
time = 2 ms.
%7 = Mod(-2*w - 3/2, w^2 - 2)
gp>  contfracpnqn([40,1,1,c])
time = 0 ms.
%8 =
[81*c + 41 81]

[2*c + 1 2]

gp>  c-(81*c+41)/(2*c+1)
time = 0 ms.
%9 = (2*c^2 - 80*c - 41)/(2*c + 1)
gp>  contfracpnqn([-41,2,c])
time = 0 ms.
%10 =
[-81*c - 41 -81]

[2*c + 1 2]

gp>  subst((-81*c-41)/(2*c+1),c,(40+29*r2)/2)
time = 2 ms.
%11 = Mod(-29/2*w - 20, w^2 - 2)
gp>  contfracpnqn([4,1,d])
time = 0 ms.
%12 =
[5*d + 4 5]

[d + 1 1]

gp>  d-(5*d + 4)/(d+1)
time = 3 ms.
%13 = (d^2 - 4*d - 4)/(d + 1)
gp>  contfracpnqn([-3,c])
time = 0 ms.
%14 =
[-3*c + 1 -3]

[c 1]

gp>  subst((-3*c + 1)/c,c,2+2*r2)
time = 1 ms.
%15 = Mod(1/2*w - 7/2, w^2 - 2)