Points on an elliptic curve over an extension field lie in the wrong point homset

[pjbruin]

In Sagemath 10.3.beta8:

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2 + x - 24)
sage: E = EllipticCurve('37a')
sage: X = E(K)
sage: P = X([3, a])
sage: P in X
False
sage: P.parent() == X
False

This is related to the fact that K is not the base field QQ of the curve, and we do get the correct output for points over QQ:

sage: Q = E(QQ)([0, 0])
sage: Q in E(QQ)
True

[pjbruin]

There is a difference in the codomains of X and P:

sage: X.codomain()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P.codomain()
Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in a with defining polynomial x^2 + x - 24

There is also a more subtle difference in the domains:

sage: X.domain()
Spectrum of Number Field in a with defining polynomial x^2 + x - 24
sage: P.domain()
Spectrum of Number Field in a with defining polynomial x^2 + x - 24
sage: X.domain() == P.domain()
False

The output False is explained by a difference in base rings:

sage: X.domain().base_ring()
Rational Field
sage: P.domain().base_ring()
Number Field in a with defining polynomial x^2 + x - 24