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read("400k-3.gp");
/* Testing all ECs with a 3-torsion point, conductor < 4*10^5 and rank > 0 */
/* Taken from LMFBD */
S = List();
{
for( i = 1, length( data ),
if( i % 100 == 1, print( "Tested ", i-1, " curves." ); );
coeffs = data[i];
E = ellinit( coeffs );
f3 = elldivpol( E, 3 ); /* 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( f3, eqn ), y, x ); /* Keep x for consistency */
K3 = nfinit( polredbest( nfsplitting( res, 6 ) ) );
E_ext = ellinit( coeffs, K3 );
Egens = ellgenerators(E);
for( j = 1, length( Egens ),
P = Egens[j];
if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
cremona_label = ellidentify(E)[1][1];
listput( S, [cremona_label, P, Q] );
write( "pari-output-ext", cremona_label );
write( "pari-output-ext", P );
write( "pari-output-ext", Q );
print( "Found EC ", cremona_label, " with point ", P );
/* 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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