///////////////////////////////////////////////////////////////////////////
//
// 		"An algorithm for determining torsion growth of elliptic curves"
// 		Enrique González-Jiménez & Filip Najman
//
//      Magma script related to Lemma 2.6 CM case
// 		
//      file: Lemma_2_6CM.txt
//
///////////////////////////////////////////////////////////////////////////
// 25/3/2019 - Magma 2.24
//////////////////////////////////////////////////////////////////////////

jCM:=[0,1728,2^4*3^3*5^3,2^3*3^3*11^3,-3^3*5^3,3^3*5^3*17^3,2^6*5^3,-2^(15)*3*5^3,-2^(15),-2^(15)*3^3,-2^(18)*3^3*5^3,-2^(15)*3^3*5^3*11^3,-2^(18)*3^3*5^3*23^3*29^3];

D:=6*21;
L:=<>;
for j in jCM do 
		E:=MinimalQuadraticTwist(EllipticCurveFromjInvariant(j));
		fac:=Factorization(DivisionPolynomial(E,81) div DivisionPolynomial(E,27):DegreeLimit:=D);
		deg81:=[Degree(g[1]): g in fac];
		printf "j=%o -- %o\n",j,deg81;
		L:=Append(L,<j,deg81>);
end for;

assert L eq <<0, []>, <1728, []>, <54000, []>, <287496, []>, <-3375, []>, <16581375, []>, 
<8000, [ 54, 108 ]>, <-12288000, [ 81 ]>, <-32768, [ 54, 108 ]>, <-884736, []>, 
<-884736000, []>, <-147197952000, []>, <-262537412640768000, []>>;

print "Done";
quit;