## [2003.11.24]y^2=x^3+7823$B$NM-M}E@(B

### y^2=x^3+7823$B$NM-M}E@$r7W;;$9$k%W%m%0%i%(B(pari/GP) y^2=x^3+7823$B$N(B4-covering$B$NM-M}E@$rC5$9%W%m%0%i%(B(C) $B"#BJ1_6J@~(B
E: y2=x3+7823
$B$NM-M}E@$r5a$a$k!#(B $B$3$3$G!"(B7823$B$OAG?t$G$"$k!#(B

$B"#BJ1_6J@~(BE$B$NH=JL<0(B $B&$(B,j-$BITJQNL(B j,$BF3$l!"(B
$B&$(B = -26438110128
j = 0
N = 26438110128
$B$G$"$k!#(B
gp>  e=ellinit([0,0,0,0,7823])
time = 38 ms.
%28 = [0, 0, 0, 0, 7823, 0, 0, 31292, 0, 0, -6759072, -26438110128, 0, [-19.85139861599267338366739366, 9.925699307996336691833696830 - 17.19181550210090147883446796*I, 9.925699307996336691833696830 + 17.19181550210090147883446796*I]~, 0.9441263528588053501078812122, -0.4720631764294026750539406061 + 0.2725458019860254346787548633*I, -5.763421470257824514359264813 + 0.E-28*I, 2.881710735128912257179632406 - 4.991269405959935527809297249*I, 0.2573176740160443415175089611]
gp>  e.j
time = 0 ms.
%29 = 0
gp>  e.disc
time = 0 ms.
%30 = -26438110128
gp>  ellglobalred(e)
time = 3 ms.
%31 = [26438110128, [1, 0, 0, 0], 1]


$B"#BJ1_6J@~(BE$B$O!"<+L@$J$M$8$lE@(BO$B$r;}$D!#(B
$B$D$^$j!"(BE$B$N$M$8$lE@72(BE(Q)tors$B$O!"(B{ O }$B$KF17?$G$"$k!#(B
gp>  elltors(e,1)
time = 68 ms.
%33 = [1, [], []]


$B"#BJ1_6J@~(BE$B$NM-M}E@72(BE(Q)$B$N(Brank$B$O!"(BCremona$B$N(Bmwrank3$B$K$h$C$F!"(B0$B$^$?$O(B1$B$G$"$k$3$H$,J,$+$k!#(B bash-2.05a$ 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: [0,0,0,0,7823]

Curve [0,0,0,0,7823] :  No points of order 2
Basic pair: I=0, J=-211221
disc=-44614310841
Two (I,J) pairs
Looking for quartics with I = 0, J = -211221
Looking for Type 3 quartics:
Trying positive a from 1 up to 17 (square a first...)
Trying positive a from 1 up to 17 (...then non-square a)
Trying negative a from -1 down to -11
Finished looking for Type 3 quartics.
Looking for quartics with I = 0, J = -13518144
Looking for Type 3 quartics:
Trying positive a from 1 up to 68 (square a first...)
Trying positive a from 1 up to 68 (...then non-square a)
(30,-12,48,116,-18)     --nontrivial...locally soluble...no rational point found (limit 10) --new (B) #1
(41,-16,-6,112,-11)     --nontrivial...--equivalent to (B) #1
Trying negative a from -1 down to -45
(-11,-20,408,1784,2072) --nontrivial...--equivalent to (B) #1
(-18,-28,312,996,838)   --nontrivial...--equivalent to (B) #1
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 0
Selmer  rank contribution from B=im(eps) = 1
Sha     rank contribution from B=im(eps) = 1
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0

Warning: Selmer rank = 1 and program finds
lower bound for rank = 0 which differs by an odd
integer from the Selmer rank.   Hence the rank must be 1 more
than reported here.  Try rerunning with a higher bound for
quartic point search.

Summary of results (all should be powers of 2):

n0 = #E(Q)[2]    = 1
n1 = #E(Q)/2E(Q) >= 1
n2 = #S^(2)(E/Q) = 2
#III(E/Q)[2]     <= 2

0 <= rank <= selmer-rank = 1

0 <= rank <= selmer-rank = 1
After descent, rank of points found is 0

The rank has not been completely determined,
only a lower bound of 0 and an upper bound of 1.
Standard parity conjectures would increase the lower bound by 1 to 1,
implying that the rank was exactly 1.
(13.2 seconds)
Enter curve: [0,0,0,0,0]

bash-2.05a$ $B;29MJ88%(B[4],[5]$B$K$h$k$H!"(Brank(E)=1$B$G$"$j!"(B E(Q)$B$N@8@.85$O!"(B
P(2263582143321421502100209233517777/119816734100955612,
-186398152584623305624837551485596770028144776655756/119816734100955613)
$B$G$"$k$3$H$,<($5$l$F$k!#(B $B"#$3$3$G$O!"(B[4],[5]$B$NJ}K!(B(4-descent)$B$G!"(BE(Q)$B$NM-M}E@$r7W;;$9$k!#(B mwrank3(2-descent)$B$N7k2L$h$j!"BJ1_6J@~(BE$B$N(B2-covering D: y2 = -18x4+116x3+48x2-12x+30 $B$rF@$k!#BJ1_6J@~(BD$B$O!"(Blocally solutable$B$G$"$k!#(B
syzygy$B$h$j!"BJ1_6J@~(BD$B$+$i!"BJ1_6J@~(BE$B$X$NM-M}JQ49&N(B:D$B"*(BE$B$r5a$a$k!#(B
$B&N(B(x,y) = ({3g4(x,1)}/{12y2},{27g6(x,1)/{12y3}) $B$?$@$7!"(B g4(X,Y) = 47280X4 + 17088X3Y + 43488X2Y2 - 85824XY3 - 11088Y4 g6(X,Y) = 1930688X6 + 2034816X5Y - 3520320X4Y2 + 8125440X3Y3 + 2998080X2Y4 + 1224576XY5 - 902592Y6 $B$G$"$k!#$h$C$F!"&N(B(x,y)=(X,Y)$B$H$9$k$H!"(B X = {985x4 + 356x3 + 906x2 - 1788x - 231}/{y2}, Y = {30167x6 + 31794x5 - 55005x4 + 126960x3 + 46845x2 + 19134x - 14103}/{y3} $B$G$"$k!#(B gp> read("stoll.gp") time = 57 ms. gp> gsi(x,y) time = 39 ms. %1 = [985/y^2*x^4 + 356/y^2*x^3 + 906/y^2*x^2 - 1788/y^2*x - 231/y^2, 30167/y^3*x^6 + 31794/y^3*x^5 - 55005/y^3*x^4 + 126960/y^3*x^3 + 46845/y^3*x^2 + 19134/y^3*x - 14103/y^3] gp> %1[1]*y^2 time = 13 ms. %2 = 985*x^4 + 356*x^3 + 906*x^2 - 1788*x - 231 gp> %1[2]*y^3 time = 15 ms. %3 = 30167*x^6 + 31794*x^5 - 55005*x^4 + 126960*x^3 + 46845*x^2 + 19134*x - 14103  (x,y)$B$rBJ1_6J@~(BD$B>e$NM-M}E@$H$9$k$H!"(B X3+7823-Y2 = 7823(324x8-4176x7+11728x6+11568x5+(-18y2-1560)x4+(116y2+5808)x3+(48y2+3024)x2+(-12y2-720)x+y4+30y2+900)(18x4-116x3-48x2+12x+y2-30)/{y6} = 0 $B$J$N$G!"(B(X,Y)$B$OBJ1_6J@~(BE$B>e$NM-M}E@$G$"$k$3$H$,J,$+$k!#(B [asir$B$K$h$k7W;;(B] [0] X=(985*x^4 + 356*x^3 + 906*x^2 - 1788*x - 231)/y^2; (985*x^4+356*x^3+906*x^2-1788*x-231)/(y^2) [1] Y=(30167*x^6 + 31794*x^5 - 55005*x^4 + 126960*x^3 + 46845*x^2 + 19134*x - 14103)/y^3; (30167*x^6+31794*x^5-55005*x^4+126960*x^3+46845*x^2+19134*x-14103)/(y^3) [2] F=red((X^3+7823-Y^2)*y^6); 45623736*x^12-882058896*x^11+5319389664*x^10-7415327824*x^9-15589236312*x^8-29289312*x^7-5925453120*x^6-7887837024*x^5+555933672*x^4-1625556816*x^3-1115246880*x^2+253465200*x+7823*y^6-211221000 [3] fctr(F); [[7823,1],[324*x^8-4176*x^7+11728*x^6+11568*x^5+(-18*y^2-1560)*x^4+(116*y^2+5808)*x^3+(48*y^2+3024)*x^2+(-12*y^2-720)*x+y^4+30*y^2+900,1],[18*x^4-116*x^3-48*x^2+12*x+y^2-30,1]]  $B"#B?9<0(Bg(x)=-18x4+116x3+48x2-12x+30$B$O!"$=$NH=JL<0$OIi$G$"$j!"#28D$N gp> f(x,0) time = 1 ms. %5 = -18*x^4 + 116*x^3 + 48*x^2 - 12*x + 30 gp> poldisc(-18*x^4+116*x^3+48*x^2-12*x+30) time = 8 ms. %6 = -6768156192768 gp> polroots(-18*x^4 + 116*x^3 + 48*x^2 - 12*x + 30) time = 286 ms. %7 = [-0.8201001084511361836541385020 + 0.E-28*I, 6.826037817661110050780091244 + 0.E-28*I, 0.2192533676172352886592458509 - 0.4996513902612574259040823588*I, 0.2192533676172352886592458509 + 0.4996513902612574259040823587*I]~  $B"#BJ1_6J@~(BE$B$N(B4-covering C$B$,0J2<$N$h$&$K7W;;$G$-$k!#(B $B6J@~(BC$B$O!"(BP3$B$N0J2<$N#2$D$N#2 x12+4x1x2-2x1x3-2x1x4-2x22-3x32+4x3x4+x42 = 0, x12-6x1x4+2x22+4x2x3+3x32+2x3x4+x42 = 0 $B6J@~(BC$B$NM-M}E@(B(x1:x2:x3:x4) \in P3$B$r(Bmax{|x1|,|x2|,|x3|,|x4|}<= 1000$B$NHO0O$GC5$9$H!"(B(681:-116:-125:142)$B$,8+$D$+$k!#(B

[C$B%W%m%0%i%$K$h$k7W;;(B]
bash-2.05a$gcc -m486 -O2 -lm x7823.c -o x7823 bash-2.05a$ ./x7823 1000 1000 1000
find4 -- a1=1000, a2=1000, a3=1000
[681,-116,-125,142]

real    23m25.373s
user    22m50.790s
sys     0m0.270s
bash-2.05a$ $B"#6J@~(BC$B$+$i6J@~(BD$B$X$NM-M}JQ49&W$r5a$a$k!#(B x=[x1; x2; x3; x4], xt=[x1,x2,x3,x4], M1=[1,0,0,-3; 0,2,2,0; 0,2,3,1; -3,0,1,1], M2=[1,2,-1,-1; 2,-2,0,0; -1,0,-3,2; -1,0,2,1], $B$H$9$k$H!"6J@~(BC$B$O!"0J2<$N$h$&$KI=8=$G$-$k!#(B xtM1x=0, xtM2x=0 M1,M2$B$N(Badjoint$B$r$=$l$>$l(BM1',M2'$B$H$9$k$H!"(B
det(x*M1+M2)=g(x),
$B$G$"$k!#$?$@$7!"(B D1=[-146,88,166,66; 88,-20,-56,60; 166,-56,-140,48; 66,60,48,-102] D2=[282,84,8,-202; 84,-84,-160,68; 8,-160,-176,158; -202,68,158,174] $B$G$"$k!#(B F1=xtD1x, F2=xtD2x $B$H$9$k$H!"(Bsyzygy
$B$rF@$k!#$h$C$F!"&W(B:C$B"*(BD $B&W(B(x)=({-F1(x)}/{F2(x)},{G(x)}/{F2(x)2})
$B$D$^$j!"(B
$B&W(B(x1:x2:x3:x4)=({146x12-176x1x2 -332x1x3 - 132x1x4 + 20x22 + 112x2x3 - 120x2x4+ 140x32 - 96x3x4+102x42}/{282x12 + 168x1x2 + 16x1x3 - 404x1x4 -84x22 -320x2x3 + 136x2x4-176x32 + 316x3x4 + 174x42}, {G(x)}/{(282x12 + 168x1x2 + 16x1x3 - 404x1x4 -84x22 -320x2x3 + 136x2x4-176x32 + 316x3x4 + 174x42)2}) $B$H$9$k!#(B $B"#6J@~(BC$B$NM-M}E@(BP(681:-116:-125:142)$B$r&W$G gp> read("stoll.gp") time = 200 ms. %1 = 0 gp> P time = 0 ms. %2 = [53463613/32109353, 23963346820191122/1031010550078609] gp> gsi(53463613/32109353, 23963346820191122/1031010550078609) time = 18 ms. %3 = [2263582143321421502100209233517777/143560497706190989485475151904721, 186398152584623305624837551485596770028144776655756/1720094998106353355821008525938727950159777043481] gp> ec(2263582143321421502100209233517777/14356049770619098948547515194721, 186398152584623305624837551485596770028144776655756/1720094998106353355821008525938727950159777043481) time = 2 ms. %4 = 0  $B$h$C$F!"BJ1_6J@~(BE$B$NM-M}E@(B R(2263582143321421502100209233517777/14356049770619098948547515194721, 186398152584623305624837551485596770028144776655756/1720094998106353355821008525938727950159777043481) $B$,F@$i$l$?!#(B
[2016.06.19$BDI5-(B] $BBJ1_6J@~(BE$B$O!"(Banalyticrank(E)=1$B$G$"$k$3$H$+$i!"(Brank(E)=1$B$,J,$+$j!"$=$N(BHeegner$BE@$r7W;;$9$k$3$H$K$h$j!"M-M}E@$r5a$a$k$3$H$,$G$-$k!#(B
Windows 10(x64)$B>e$N(Bpari/gp-2.7.5(x64)$B$K$h$j!"BJ1_6J@~(BE$B$N(BHeegner$BE@$r7W;;$9$k$H!"0J2<$N$h$&$K$J$k!#(B Reading GPRC: gprc.txt ...Done. GP/PARI CALCULATOR Version 2.7.5 (released) amd64 running mingw (x86-64/GMP-6.0.0 kernel) 64-bit version compiled: Oct 26 2015, gcc version 4.9.3 (GCC) threading engine: single (readline v6.2 enabled, extended help enabled) Copyright (C) 2000-2015 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 512000000, primelimit = 500000 (10:25) gp > e=ellinit([0,0,0,0,7823]) time = 1 ms. %1 = [0, 0, 0, 0, 7823, 0, 0, 31292, 0, 0, -6759072, -26438110128, 0, Vecsmall([1]), [Vecsmall([128, -1])], [0, 0, 0, 0, 0, 0, 0, 0]] (10:26) gp > elltors(e,1) time = 2 ms. %2 = [1, [], []] (10:26) gp > ellanalyticrank(e) time = 3min, 9,311 ms. %3 = [1, 73.280985665204125605489526771501344538] (10:29) gp > P=ellheegner(e) (10:29) gp > P=ellheegner(e) time = 11h, 46min, 44,151 ms. %4 = [2263582143321421502100209233517777/143560497706190989485475151904721, 186398152584623305624837551485596770028144776655756/1720094998106353355821008525938727950159777043481] (22:17) gp > ellisoncurve(e,P) %5 = 1 (22:19) gp > ellheight(e,P) %6 = 77.617773768638083972811277401165589664  $B$3$N$h$&$K!"BJ1_6J@~(BE$B$NM-M}E@(B (2263582143321421502100209233517777/143560497706190989485475151904721, 186398152584623305624837551485596770028144776655756/1720094998106353355821008525938727950159777043481) height 77.617773768638083 $B$,7W;;$G$-$?!#(B