#8319: elliptic curve canonical height bug for non-minimal models
-------------------------------+--------------------------------------------
Reporter: cremona | Owner: cremona
Type: defect | Status: new
Priority: major | Milestone:
Component: elliptic curves | Keywords: canonical height
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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For canonical heights of points on elliptic curves defined over QQ, we
call the pari function ellheight(), BUT that function only gives the
correct value for global minimal models! (At primes where the model is
not minimal the local component is wrong).
Here is an example to show this:
{{{
sage: E = EllipticCurve([-5580472329446114952805505804593498080000,
....:
-157339733785368110382973689903536054787700497223306368000000])
sage:
P3=E([204885147732879546487576840131729064308289385547094673627174585676211859152978311600/23625501907057948132262217188983681204856907657753178415430361,92736254270288706000052529616433462503893110244786566575440613880920567197984949809570153263207165494624214991751142500000000/114834283590033957142081201488956527887145361353994063932282392800295014255070987824900081891])
sage: P3.height()
157.086024926474
sage: 4*(P3.height())-(2*P3).height() # should == 0
-1.38629436111989
}}}
while on the minimal model:
{{{
sage: Emin = E.minimal_model()
sage: urst = E.isomorphism_to(Emin)
sage: 4*urst(P3).height()-urst(2*P3).height()
0.000000000000000
}}}
The cure is to compute the minimal model and transfer the point there
before computing the height, as illustrated above. (Of course, pari could
do that too, but this behaviour has been tolerated by pari users for many
years!)
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8319>
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