#15483: Simon 2-descent gives RuntimeError for an elliptic curve over a
quadratic
field
-------------------------------------------------+-------------------------
Reporter: pbruin | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.2
Component: elliptic curves | Resolution:
Keywords: simon_two_descent | Merged in:
Authors: | Reviewers:
Report Upstream: Not yet reported upstream; | Work issues:
Will do shortly. | Commit:
Branch: | Stopgaps:
Dependencies: #11005 |
-------------------------------------------------+-------------------------
Old description:
> [See #15608 for a list of open simon_two_descent tickets]
>
> The following tries to do 2-descent on an elliptic curve over a quadratic
> field:
> {{{
> K.<s>=QuadraticField(229)
> c4=2173-235*(1-s)/2
> c6=-124369+15988*(1-s)/2
> E=EllipticCurve([-c4/48,-c6/864])
> E.simon_two_descent()
> }}}
> It fails with a `RuntimeError`:
> {{{
> Traceback (most recent call last):
> ...
> RuntimeError:
> *** at top-level: ans=bnfellrank(K,[0,0,0,
> *** ^--------------------
> *** in function bnfellrank: ...eqtheta,rnfeq,bbnf];rang=
> *** bnfell2descent_gen(b
> *** ^--------------------
> *** in function bnfell2descent_gen: ...riv,r=nfsqrt(nf,norm(zc))
> *** [1];if(DEBUGLEVEL_el
> *** ^--------------------
> *** array index (1) out of allowed range [none].
> An error occurred while running Simon's 2-descent program
> }}}
> It appears that a certain number field element is expected to be a
> square, but in fact isn't.
>
> The bug is not fixed by first changing to an integral model.
>
> See #11041 for other bug reports related to Denis Simon's script, some of
> which may be caused by the fact that Sage currently uses an outdated
> version.
New description:
[See #15608 for a list of open simon_two_descent tickets]
The following tries to do 2-descent on an elliptic curve over a quadratic
field:
{{{
K.<s>=QuadraticField(229)
c4=2173-235*(1-s)/2
c6=-124369+15988*(1-s)/2
E=EllipticCurve([-c4/48,-c6/864])
E.simon_two_descent()
}}}
It fails with a `RuntimeError`:
{{{
Traceback (most recent call last):
...
RuntimeError:
*** at top-level: ans=bnfellrank(K,[0,0,0,
*** ^--------------------
*** in function bnfellrank: ...eqtheta,rnfeq,bbnf];rang=
*** bnfell2descent_gen(b
*** ^--------------------
*** in function bnfell2descent_gen: ...riv,r=nfsqrt(nf,norm(zc))
*** [1];if(DEBUGLEVEL_el
*** ^--------------------
*** array index (1) out of allowed range [none].
An error occurred while running Simon's 2-descent program
}}}
The problem is apparently that the square root of `norm(zc)` (which is
indeed a square) is not computed due to a precision bug; see comment:7
below.
--
Comment (by pbruin):
It seems that the bug is caused by a precision problem in `subst`, called
by `nfrealsign` in `ell.gp`. The following GP input demonstrates this
(using the latest version of `ell.gp`, see #11005):
{{{
\r src/ext/pari/simon/ell.gp
b =
-1554544300737274875964190134520312870631312460283689944298138572669148295776039072867720281361776956435252620954745928376624817557704277432961924925312*y
+
23524523971732905757341977352314040726186200302188191824300117738073539522011689544444863977622786771332621915440577829842674416407299864303146477224320
a = Mod(b, y^2 - 229);
\p 38 \\ default precision
K = bnfinit(y^2 - 229);
nfrealsign(K, a, 1) \\ result: 1
nfrealsign(K, a, 2) \\ result: -1 (wrong)
subst(b, y, K.roots[2]) \\ result: -7.695704335233296721 E112, incorrect
to this precision
\p 500 \\ increase precision
K = bnfinit(y^2 - 229);
nfrealsign(K, a, 1) \\ result: 1
nfrealsign(K, a, 2) \\ result: 1
subst(b, y, K.roots[2]) \\ result:
55550556624985845118007242443189926820306719407.796355955086894963616120700422308457186069825115395206111444495243228840597412527828358943327267422898011087182852030228685210492253493260963313348652457717717703991690402543456279919008587884729307897541432624377818611055431792706380900043638904108307170236335925883131494247446074384269328376967381902948861789570537169455258630011202296209679102466188417
}}}
The value of `K.roots` does not seem to be the problem; the roots are
calculated correctly to the requested precision.
--
Ticket URL: <http://trac.sagemath.org/ticket/15483#comment:7>
Sage <http://www.sagemath.org>
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