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/* Candidate counterex mod3 not surj, unexpected density, rank >=1 */
label = [ "220a3", "220a4", "660c3", "660c4", "770f3", "770f4" ];
{
for( i = 1, length( data ),
print( "Testing curve ", i, ", label ", label[i] );
E = ellinit( label[i] );
f9 = elldivpol( E, 9 ); /* 3-division polynomial */
eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
res = substpol( polresultant( f9, eqn ), y, x ); /* Keep x for consistency */
print( "Computing K9..." );
K9 = nfinit( nfsplitting( res, 3^5*2 ) ); /* mo3 is at most 6 */
print( "Computed!" );
E_ext = ellinit( E[1..5], K9 );
Egens = ellgenerators(E);
for( j = 1, length( Egens ),
P = Egens[j];
if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
print( "Found EC ", cremona_label, " with point ", P,
" and 3-division point ", Q, " over K9" );
/* Check for 3-divisibility up to torsion */
/* T = elltors(E)[3][1];
if( ellisdivisible( E, elladd(E,P,T), 3 ),
print( "P+T_1 is 3-divisible!" );
);
T = ellmul(E,T,2);
if( ellisdivisible( E, elladd(E,P,T), 3 ),
print( "P+T_2 is 3-divisible!" );
); */
);
);
);
}
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