## [2002.06.30]4-descent,8-descent

### 4-descent\$B\$K\$h\$C\$FM-M}E@\$rC5\$9%W%m%0%i%`(B(pari/gp) 8-descent\$B\$K\$h\$C\$FM-M}E@\$rC5\$9%W%m%0%i%`(B(pari/gp)

\$B"#<+A3?t(Bn\$B\$,9gF1?t\$G\$"\$k\$3\$H\$r<(\$9\$K\$O!"BJ1_6J@~(B
En: y2 = x3-n2x
\$B\$N<+L@\$G\$J\$\$(B(y!=0\$B\$G\$"\$k(B)\$BM-M}E@\$r>/\$J\$/\$H\$b#1\$D5a\$a\$k\$3\$H\$,\$G\$-\$l\$P!"==J,\$G\$"\$k!#(B

\$BF1MM\$K!"(Bn\$B\$,(B(3/\$B&P(B)-\$B9gF1?t\$G\$"\$k\$3\$H\$r<(\$9\$K\$O!"BJ1_6J@~(B
Cn: y2 = x3+2nx2-3n2x
\$B\$N<+L@\$G\$J\$\$(B(y!=0\$B\$G\$"\$k(B)\$BM-M}E@\$r>/\$J\$/\$H\$b#1\$D5a\$a\$k\$3\$H\$,\$G\$-\$l\$P!"==J,\$G\$"\$k!#(B
\$B\$h\$C\$F!"(BEn\$B\$^\$?\$O(BCn\$B\$N(BMordell-Weil rank\$B\$,#10J>e\$G\$"\$k\$3\$H\$,J,\$+\$l\$PNI\$/!"(Brank\$B\$r@53N\$K5a\$a\$k\$3\$H\$OI,?\\$G\$J\$\$!#(B

Cremona\$B\$N(Bmwrank\$B\$G\$O!"9b\$5(B(Canonical Height)\$B\$NBg\$-\$\$M-M}E@\$r5a\$a\$k\$N\$K!";~4V\$,\$+\$+\$k\$N\$G!"(Brank\$B\$N3NDj\$r\$7\$J\$\$JL\$NJ}K!(B(4-descent,8-descent)\$B\$G!"M-M}E@\$r5a\$a\$F\$_\$k!#(B

\$B"#(BAllan J. MacLeod\$B\$NO@J8(B\$B\$K5-=R\$5\$l\$F\$\$\$k(B4-descent\$B\$N%"%k%4%j%:%`\$r>R2p\$9\$k!#(B
\$B0L?t#2\$NE@\$r>/\$J\$/\$H\$b#1\$D;}\$DBJ1_6J@~(B
E: y2 = x3+ax2+bx ---------- (1)
\$B\$r9M\$(\$k!#(B
\$B0lHL@-\$r<:\$&\$3\$H\$J\$/!"(Bx=du2/v2, y=duw/v3, \$B\$@\$@\$7!"(Bd\$B\$O(Bsquarefree(\$B#2>h0x;R\$r4^\$^\$J\$\$(B), (d,v)=1 \$B\$HCV\$/\$3\$H\$,\$G\$-\$k!#(B
(1)\$B\$KBeF~\$9\$k\$H!"(B
d2w2 = d3u4+d2au2b2+dbv4 ---------- (2)
\$B\$H\$J\$k!#\$h\$C\$F!"(Bd|b, b=de\$B\$G\$"\$k!#(B
d\$B\$O(Bb\$B\$N(Bsquarefree\$B0x;R\$G\$"\$j!"M-8B8D\$7\$+\$J\$\$!#(Bd\$B\$OIi\$NCM\$b \$B:G=i\$K>/\$7C1=c\$J#2     h2 = df2+afg+eg2 ---------- (3)
\$B\$r9M\$(\$k!#(B
\$B#20,f0,g0)\$B\$G(Bg0!=0\$B\$J\$k\$b\$N\$r8+\$D\$1\$k\$3\$H\$,\$G\$-\$?\$H\$9\$k!#(B
\$B\$3\$N\$H\$-!"(B(3)\$B\$N2r\$r     y2 = dx2+ax+e
\$B\$H\$J\$k!#E@(B(x0,y0)=(f0/g0,h0/g0)\$B\$rDL\$k79\$-(Bm\$B\$ND>@~\$,6J@~(B(3)\$B\$H:F\$S=P9g\$&E@\$N(Bx\$B:BI8\$O!"(B
x = {a+dx0+m(mx0-2y0)}/(m2-d) ----------- (4)
\$B\$G\$"\$k!#\$3\$3\$G!"(Bm=p/q\$B\$,M-M}?t\$G\$"\$k\$H\$9\$k\$H!"(B
x = {ag0q2+df0q2+p(f0p-2h0q)}/{g0(p2-dq2)} ----------- (5)
\$B\$H\$J\$k!#(B
\$B85\$NLdBj(B(2)\$B\$r2r\$/\$?\$a\$O!"(Bf/g=v2/u2(\$B!z:G=i\$N(Bx\$B\$NI=<(\$HHf3S\$9\$k\$H!"(B(u,v)\$B\$,F~\$l49\$o\$C\$F\$\$\$k\$3\$H\$KCm0U\$9\$k(B)\$B\$G%Q%i%a!<%?I=<(\$9\$k\$H!"0J2<\$N#2\$D\$N#2     ku2 = g0(p2-dq2) ----------- (6)
kv2 = f0p2-2h0pq+(ag0+df0)q2 ----------- (7)
\$B\$3\$3\$G!"(Bk\$B\$O(Bsquarefree\$B\$G\$"\$k!#(B
k\$B\$O(B(6),(7)\$B\$N1&JU\$N#20(a2-4b)\$B\$r3d\$j@Z\$k!#(Bk0\$B\$r=*7k<0\$N(Bsquarefree\$B0x;R(B(\$BIi\$NCM\$b2DG=(B)\$B\$H\$9\$k!#(B
k=k0\$B\$KBP\$7\$F!"(B(6)\$B\$N#1\$D\$N2r(B(u0,p0,q0)\$B\$r8+\$D\$1\$?\$H\$9\$k!#(B
\$B6J@~(B(6)\$B\$H(B(p0,q0)\$B\$rDL\$k79\$-(Br/s\$B\$ND>@~\$H\$N8rE@\$r9M\$(\$k\$3\$H\$K\$h\$j!"(B(p,q)\$B\$r     p = 2u0k0rs-p0(g0s2+k0r2) ----------- (8)
q = q0(g0s2-k0r2) ----------- (9)
k0*(7)\$B\$K!"(B(8),(9)\$B\$rBeF~\$9\$k\$H!"#4     z2 = z1r4+z2r3s+z3r2s2+z4r3+z5s4 ----------- (10)
\$B\$rF@\$k!#\$?\$@\$7!"(B
z1 = k03(ag0q02+df0q02+f0p02-2h0p0q0) ---------- (11)
z2 = 4u0k03(h0q0-f0p0) ---------- (12)
z3 = 2k02(f0(2u02k0-dg0q02+g0p02)-ag02q02) ---------- (13)
z4 = -4u0g0k02(f0p0+h0q0) ---------- (14)
z5 = g02k0(ag0q02+df0q02+f0p02+2h0p0q0) ---------- (15)
\$B\$G\$"\$k!#(B
\$B#42/v2\$B\$rF@\$F!"(B(2)\$B\$h\$j!"(Bw=sqrt(du4+au2v2+ev4), y=duw/v3\$B\$rF@\$k!#(B

\$B"#F1MM\$K(B\$B\$K5-=R\$5\$l\$F\$\$\$k(B8-descent\$B\$N%"%k%4%j%:%`\$r>R2p\$9\$k!#(B
4-descent\$B\$G\$O!"#4 \$B\$=\$3\$G!"(B(10)\$B\$N1&JU\$,(B(r,s)\$B\$N#2Dj\$9\$k!#\$D\$^\$j!"(B
z1r4+z2r3s+z3r2s2+z4r3+z5s4 = (u1r2+u2rs+u3s2)(v1r2+v2rs+v3s2)----------- (17)
\$B\$H\$9\$k!#\$3\$3\$G!"(Bu1,u2,u3,v1,v2,v3\$B\$OM-M}@0?t\$G\$"\$j!"0J2<\$rK~\$?\$9!#(B
z1 = u1v1
z5 = u3v3
z2 = u1v2+v1u2
z4 = u3v2+v3u2
z3 = u1v3+u2v2+u3v1

\$B     u1r2+u2rs+u3s2 = k1m2 ----------- (18)
v1r2+v2rs+v3s2 = k1t2 ----------- (19)
\$B\$3\$3\$G!"(Bk1\$B\$O(Bsquarefree\$B\$G\$"\$k!#(Bk1\$B\$O!"(B(18),(19)\$B\$N:8JU\$N(B \$B#2     u12u32-u1u2v2v3-2u1u3v1v3+u1u3v22+u22v1v3-u2u311v2+u32v12 ---------- (20)
\$B\$r3d\$j@Z\$k!#(B
\$B#21,t1,r1,s1)\$B\$r8+\$D\$1\$k\$3\$H\$,\$G\$-\$?\$i!"(B(r,s)\$B\$r0J2<\$N\$h\$&\$K!"%Q%i%a!<%?I=<(\$G\$-\$k!#(B
r = i2k1r1-2ijk1t1+j2(r1v1+s1v2) ---------- (21)
s = s1(i2k1-j2v1) ---------- (22)
k1*(18)\$B\$r(B(21),(22)\$B\$GCV\$-49\$(\$k\$H!"0J2<\$N#4     c1i4+c2i3j+c3i2j2+c4ij3+c5j4 ------------ (23)
\$B\$?\$@\$7!"(B
c1 = k13(r12u1+r1s1u2+s12u3) ---------- (24)
c2 = -2k13t1(2r1u1+s1u2) ---------- (25)
c3 = k12(4k1t12u1+2r12u1v1+2r1s1u1v2+s12(u2v2-2u3v1)) ---------- (26)
c4 = -2k12t1(2r1u1v1+s1(2u1v2-u2v1)) ---------- (27)
c5 = k1(r12u1v12+r1s1v1(2u1v2-u2v1)+s12(u1v22-v1(u2v2-u3v1))) ---------- (28)
\$B\$G\$"\$k!#(B
\$B#4

\$B"#(BAllan J. MaxLeod\$B\$NO@J8(B\$B\$G\$O!"(BUBASIC\$B\$N%W%m%0%i%`\$r%3%s%Q%\$%k\$G\$-\$k9b5i8@8l\$KJQ49\$7\$F!"(B200MHz\$B\$^\$?\$O(B300MHz PC\$B\$G \$B\$3\$3\$G\$O!"(B4-descent, 8-descent\$B\$N%"%k%4%j%:%`\$r(Bpari/GP\$B\$G%W%m%0%i%`\$7\$F\$_\$k!#(B
\$B:#2s\$h\$j!"(Bpari-2.1.4\$B\$*\$h\$S(B(gp\$B%3%s%Q%\$%i\$G\$"\$k(B)gp2c-0.0.1pl1\$B\$r;HMQ\$7\$F\$_\$?!#(B
4-descent,8-descent\$B\$K\$h\$C\$F!"(BLoox T9/80W(Crusoe 5800/800MHz)\$B>e\$G!"(B7\$B\$,9gF1?t\$G\$"\$k\$3\$H\$r3NG'\$9\$k!#(B
(1)4-descent
```bash-2.05a\$ gp2c-run -g 4-descent.gp

GP/PARI CALCULATOR Version 2.1.4 (released)
i386 running netbsd 32-bit version
(readline v4.2a enabled, extended help available)

Copyright (C) 2002 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.

realprecision = 28 significant digits
seriesprecision = 16 significant terms
format = g0.28

parisize = 4000000, primelimit = 500000
gp> congr(7,9,9,9)
[x,y]=[112/9, 980/27]
time = 113 ms.
%1 = 1
gp> \q
Good bye!
```
(2)8-descent
```bash-2.05a\$ gp2c-run -g 8-descent.gp

GP/PARI CALCULATOR Version 2.1.4 (released)
i386 running netbsd 32-bit version
(readline v4.2a enabled, extended help available)

Copyright (C) 2002 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.

realprecision = 28 significant digits
seriesprecision = 16 significant terms
format = g0.28

parisize = 4000000, primelimit = 500000
gp> congr(7,9,9,9,9)
[x,y]=[-49/25, -1176/125]
time = 260 ms.
%1 = 1
gp> \q
Good bye!
```

### [\$B;29MJ88%(B]

• Allan J. MacLeod, "A simple practical higher descent for large height rational points on certain elliptic curves", Dept. of Mathmatics and Statics Univercity of Paisley, June 15, 2002.
• Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.
• Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", GTM 151, Springer-Verlag New York Inc., 1994, ISBN0-387-94328-5.

 Last Update: 2005.06.12 H.Nakao