#19689: Scaling of Weierstrass equations by units, over number fields
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Reporter: cremona | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.10
Component: elliptic curves | Resolution:
Keywords: reduction scaling Weierstrass | Merged in:
model | Reviewers:
Authors: John Cremona | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: |
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Description changed by cremona:
Old description:
> We already have (semi-)global minimal models for elliptic curves over
> number fields in #18662 (but see also #19665), and at that time I
> implemented reduction via scaling by units over real quadratic fields
> only. Here I extend that to arbitrary number fields. The method is
> similar to that used in Magma, though only implemented there currently
> for totally real fields.
>
> The idea is to map (c4,c6) into R^{r1+r2} via the vector indexed by
> infinite places v whose coordinates are
> d_v*log(|c4|_v^(1/4)+|c6|_v^(1/6)), and reduce this vector modulo
> translation by the lattice which is the image of the unit group.
New description:
We already have (semi-)global minimal models for elliptic curves over
number fields in #18662 (but see also #19665), and at that time I
implemented reduction via scaling by units over real quadratic fields
only. Here I extend that to arbitrary number fields. The method is
similar to that used in Magma, though only implemented there currently for
totally real fields.
The idea is to map (c4,c6) into {{{R^(r1+r2)}}} via the vector indexed by
infinite places v whose coordinates are
{{{d_v*log(|c4|_v^(1/4)+|c6|_v^(1/6))}}}, and reduce this vector modulo
translation by the lattice which is the image of the unit group.
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Ticket URL: <http://trac.sagemath.org/ticket/19689#comment:1>
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