On Jan 15, 2008 10:25 AM, John Cremona <[EMAIL PROTECTED]> wrote:
>
> I like Robert's suggestion. If the user wants n independent generic
> points, construct a large enough field (transcendence degree n) to
> contain them.
>
> A useful change Magma made relatively recently (a couple of years or
> so ago) was to aloow points on an elliptic curve to have coordinates
> in an extension of the base field of the curve -- as one would when
> working mathematically. e.g. given a curve defined over QQ you can
> define points on E(K) for e.g. K=a number field, or K=a function field
> (such as the function field of E, to get a generic point). Of course,
> these points "know" what their curve is so you can do point arithmetic
> on them and so on.
>
> I don't see why this should be workable in Sage too (maybe it is
> already? if so I will retire shame-faced from the discussion...)
It's sort of half-way there. You can do:
sage: K.<a> = NumberField(x^2 + x - (3^3-3))
sage: E = EllipticCurve('37a')
sage: X = E(K)
but stupidly X is wrong:
sage: X
Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x
over Rational Field
though:
sage: X.domain()
Spectrum of Number Field in a with defining polynomial x^2 + x - 24
However,
sage: P = X([3,a]);
boom with a TypeError
So this obviously needs work. In fact, this counts as a bug.
http://trac.sagemath.org/sage_trac/ticket/1785
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