id summary reporter owner description type status priority milestone component resolution keywords cc merged author reviewer upstream work_issues branch commit dependencies stopgaps
27884 Bug in quadratic twist of elliptic curves cremona cremona "If E is an elliptic curve defined over a field K and L/K is a quadratic extension then there is an elliptic curve E'/K isomorphic to E over L but not over K, called the quadratic twist of E by L/K. In case L=K(sqrt(d)) for some non-square d in K, this is also called the quadratic twist of E by d; this is always the case except in characteristic 2. When the characteristic is not 2 or 3 and E has short Weierstrass equation Y^2=X^3+aX+b then *usually* the twist by d is obtained by replacing (a,b) by (a*d^2,b*d^3), **but** there is one case for which this is incorrect, namely when d=-1 and b=0. In this case, the twist takes (a,0) to (-4*a,0). We do not implement this properly:
{{{
sage: E = EllipticCurve([0,0,0,1,0]); E
Elliptic Curve defined by y^2 = x^3 + x over Rational Field
sage: E.quadratic_twist(-1)
Elliptic Curve defined by y^2 = x^3 + x over Rational Field
}}}" defect closed trivial sage-duplicate/invalid/wontfix elliptic curves invalid quadratic twist defeo N/A