///////////////////////////////////////////////////////////////////////////
// "Growth of torsion groups of elliptic curves upon base changes"
// Enrique González-Jiménez & Filip Najman
///////////////////////////////////////////////////////////////////////////

// 13/10/2016 - Magma 2.21

// Magma script related to Theorem7.1

// It is known that there are only finitely many j-invariants
// of elliptic curves E/Q with a 14-isogeny or a 21-isogeny
// In Table 4 from Lozano-Robledo's paper we checked the following:
// * If E has a 14-isogeny, then j(E) in {-3375,16581375} 
//    and the Cremona labels of the representatives in these 
//    classes of smallest conductor are [49a1,49a3] and [49a2,49a4] resp.
// * If E has a 21-isogeny, then j(E) in {-140625/8, 3375/2, -1159088625/2097152, -189613868625/128}
//    and the Cremona labels of the representatives in these 
//    classes of smallest conductor are [162b1,162c2],[162b2,162c1],[162b3,162c4] and [162b4,162c3] resp.
// 
// Now, in the LMFDB database (www.lmfdb.org) we obtain that
// G_E(7) of the above elliptic curves belongs to
// 7B.1.5, 7B.1.2, 7B.2.1, 7B.2.3, 7B
// Since in all the above cases we have j(E)\ne 0,
// we have that if EE is an elliptic curve with a 14-isogeny or a 21-isogeny
// then \pm G_E(7)=\pm G_EE(7)

load "subgroups.m";

G1:=GL2SubgroupFromLabel("7B.1.5");
G2:=GL2SubgroupFromLabel("7B.1.2");
G3:=GL2SubgroupFromLabel("7B.2.1");
G4:=GL2SubgroupFromLabel("7B.2.3");
G5:=GL2SubgroupFromLabel("7B");

G6:=GL2SubgroupFromLabel("7B.1.3"); 

for G in [G1,G2,G3,G4,G5,G6] do
  GG:=sub<GL(2,7) | [g : g in G], [-g : g in G]>;
 printf "%o - %o\n",GL2SubgroupLabel(G), GL2SubgroupLabel(GG);
end for;

/* OUT: We check that it is not possible that an 
elliptic curve E/Q with a 14-isogeny or a 21-isogeny 
has G_E(7) = 7B.1.3

7B.1.5 - 7B.6.2
7B.1.2 - 7B.6.2
7B.2.1 - 7B
7B.2.3 - 7B
7B - 7B
7B.1.3 - 7B.6.3
*/