///////////////////////////////////////////////////////////////////////////
//
// 		"An algorithm for determining torsion growth of elliptic curves"
// 		Enrique González-Jiménez & Filip Najman
//
//      Magma script related to Lemma 2.10 and Lemma 2.11
// 		
//      file: Lemma_2_10_11.txt
//
///////////////////////////////////////////////////////////////////////////
// 27/3/2019 - Magma 2.24
///////////////////////////////////////////////////////////////////////////
// Lemma 2.10: Points of order 37

_<x>:=PolynomialRing(Rationals());
f:=x^3 - 1155*x + 16450;
E:=EllipticCurve(f);
assert CremonaReference(MinimalQuadraticTwist(E)) eq "1225h1";
assert jInvariant(E) eq -7*11^3;
g:=x^6 - 210*x^5 - 8085*x^4 + 125300*x^3 + 4251975*x^2 - 16133250*x - 408849875;
K<a>:=NumberField(g);
_<y>:=PolynomialRing(K);
p2:=y^2-(a^3 - 1155*a + 16450);
L:=AbsoluteField(ext<K|p2>);
P:=Points(BaseChange(E,L),L!a)[1];
assert Order(P) eq 37;

///////////////////////////////////////////////////////////////////////////
// Lemma 2.11: Points of order 17

_<x>:=PolynomialRing(Rationals());
f:=x^3 - 95115*x - 12657350;
E:=EllipticCurve(f);
assert CremonaReference(MinimalQuadraticTwist(E)) eq "14450n1";
assert jInvariant(E) eq -17*373^3/2^17;
g:=x^4 + 340*x^3 + 510*x^2 - 5560700*x - 237673175;
K<a>:=NumberField(g);
_<y>:=PolynomialRing(K);
p2:=y^2-(a^3 - 95115*a - 12657350);
L:=AbsoluteField(ext<K|p2>);
P:=Points(BaseChange(E,L),L!a)[1];
assert Order(P) eq 17;

print "Done";
quit;