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authorSebastiano Tronto <sebastiano@tronto.net>2026-06-14 09:58:21 +0200
committerSebastiano Tronto <sebastiano@tronto.net>2026-06-14 09:58:21 +0200
commit5ea79c7ae0d44686f1df05c4a016652afbe58968 (patch)
tree05f6a052373f5fe0942777a48303b2f8b884cbe1 /l-multiples_l-torsion/test9.gp
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1/* Candidate counterex mod3 not surj, unexpected density, rank >=1 */
2label = [ "220a3", "220a4", "660c3", "660c4", "770f3", "770f4" ];
3
4{
5for( i = 1, length( data ),
6 print( "Testing curve ", i, ", label ", label[i] );
7 E = ellinit( label[i] );
8 f9 = elldivpol( E, 9 ); /* 3-division polynomial */
9 eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
10 res = substpol( polresultant( f9, eqn ), y, x ); /* Keep x for consistency */
11 print( "Computing K9..." );
12 K9 = nfinit( nfsplitting( res, 3^5*2 ) ); /* mo3 is at most 6 */
13 print( "Computed!" );
14 E_ext = ellinit( E[1..5], K9 );
15 Egens = ellgenerators(E);
16 for( j = 1, length( Egens ),
17 P = Egens[j];
18 if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
19 print( "Found EC ", cremona_label, " with point ", P,
20 " and 3-division point ", Q, " over K9" );
21 /* Check for 3-divisibility up to torsion */
22 /* T = elltors(E)[3][1];
23 if( ellisdivisible( E, elladd(E,P,T), 3 ),
24 print( "P+T_1 is 3-divisible!" );
25 );
26 T = ellmul(E,T,2);
27 if( ellisdivisible( E, elladd(E,P,T), 3 ),
28 print( "P+T_2 is 3-divisible!" );
29 ); */
30 );
31 );
32);
33}
34

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