question of quadratic_twist of elliptic curve

asked 2014-11-03 05:30:39 -0500

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from wiki http://en.wikipedia.org/wiki/Twists_o... elliptic curve cardinality and elliptic curve'quadratic_twist cardinality

I try some short_weierstrass_model is isomorphic ,but long Weierstrass model is not isomorphic,and not equal=2*q+2.

E = EllipticCurve('11a1');E;Et = E.quadratic_twist(2);Et;Es=E.short_weierstrass_model();Es;Ess=Et.short_weierstrass_model();Ess;

SQ.<x>=QuadraticField(2);SQ;Ee11 = EllipticCurve(SQ,[-53568,-8643456]);Ee11;E211 = EllipticCurve(SQ,[-13392,-1080432]);Ess;E211;Ee11.is_isomorphic(E211);

x=var("x");K. = NumberField(x^2+x+2, 'a');K;EE = EllipticCurve(K,'11a1');E;EEE=EllipticCurve(K,[0,1,0,-41,-199]);EEE;E.is_isomorphic(EEE); m=GF(5,3,"v");m;EER = EllipticCurve(m,'11a1');EER;EEER=EllipticCurve(m,[0,1,0,-41,-199]);EEER; EER.cardinality();EEER.cardinality();7+5==2*3+2

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