On Jan 15, 2008, at 4:54 PM, Robert Bradshaw wrote:
> What about
>
> sage: K.<a> = NumberField(x^2 + x - (3^3-3))
> sage: E = EllipticCurve('37a'); E
> Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
> sage: X = E.change_ring(K); X
> Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in
> a with defining polynomial x^2 + x - 24
But I think John wants the curve to be still defined over the base
field. If E/K is an elliptic curve, and L/K is a field extension, and
P is in E(L), then you lose information if you base change the curve
up to E/L and think of P as a point on E/L. For example P doesn't
have any non-trivial galois conjugates any more. To retain these you
need to "remember" that the curve is actually defined over K.
david
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