On Jan 15, 2008, at 11:08 AM, William Stein wrote:
>
> 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
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
(Though the E(K) notation is nicer, and should work too.)
- Robert
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