#19276: precision problem computing heights on elliptic curves
-------------------------+-------------------------------------------------
Reporter: | Owner:
cremona | Status: needs_review
Type: | Milestone: sage-6.9
defect | Resolution:
Priority: major | Merged in:
Component: | Reviewers:
elliptic curves | Work issues:
Keywords: | Commit:
height precision | 3ea3fc73d85a9b50fb618f63c345bce415cf7e1a
Authors: John | Stopgaps:
Cremona |
Report Upstream: N/A |
Branch: |
u/cremona/19276 |
Dependencies: |
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Description changed by cremona:
Old description:
> Computing heights of points over number fields (this does not effect
> things over Q): in the expression (from Silverman's paper) for the
> number of terms needed for the requested precision for local height at an
> archimedean place v, there is log(max(1,1/|D|_v)) where D is the
> discriminant and v the valuation. Over Q D is integral so this term is
> always 0. Over number fields, |D|_v can be extremely small for some v,
> indistinguishable from 0 to moderate precision.
>
> Example (from an LMFDB curve):
> {{{
> sage: K.<a> = NumberField(x^2-x-104)
> sage: E = EllipticCurve([1, a - 1, 1, -816765673272*a - 7931030674178,
> 1478955604013312315*a + 14361086227143654561])
> sage: P = E(5393511/49*a + 52372721/49 , -33896210324/343*a -
> 329141996591/343 )
> sage: P.height()
> }}}
>
> I have a fix and am testing now.
New description:
Computing heights of points over number fields (this does not effect
things over Q): in the expression (from Silverman's paper) for the number
of terms needed for the requested precision for local height at an
archimedean place v, there is log(max(1,1/|D|_v)) where D is the
discriminant and v the valuation. Over Q D is integral so this term is
always 0. Over number fields, |D|_v can be extremely small for some v,
indistinguishable from 0 to moderate precision.
Example (from an LMFDB curve):
{{{
sage: K.<a> = NumberField(x^2-x-104)
sage: E = EllipticCurve([1, a - 1, 1, -816765673272*a - 7931030674178,
1478955604013312315*a + 14361086227143654561])
sage: P = E(5393511/49*a + 52372721/49 , -33896210324/343*a -
329141996591/343 )
sage: P.height()
}}}
This raises an error before the fix in this branch; after, it works fine.
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Ticket URL: <http://trac.sagemath.org/ticket/19276#comment:2>
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