#14489: _S_class_group_and_units is mathematically incorrect
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Reporter: robharron | Owner: davidloeffler
Type: defect | Status: new
Priority: critical | Milestone: sage-5.10
Component: number fields | Keywords: S-class group
Work issues: | Report Upstream: N/A
Reviewers: | Authors:
Merged in: | Dependencies:
Stopgaps: |
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The output of _S_class_group_and_units is incorrect, and hence the output
of selmer_group as well, in some cases where S contains non-principal
ideals. Here's an example:
{{{
sage: K.<a> = NumberField(x^3 - 381 * x + 127)
sage: S = tuple(K.primes_above(13))
sage: K.selmer_group(S, 2)
[-7/13*a^2 - 140/13*a + 36/13,
14/13*a^2 + 267/13*a - 85/13,
7/13*a^2 + 127/13*a - 49/13,
-1,
1/13*a^2 + 20/13*a - 7/13,
1/13*a^2 - 19/13*a + 6/13,
121,
10/13*a^2 + 44/13*a - 4555/13]
}}}
It's fairly easy, using Sage, to see that the S-2-Selmer group of K is an
8-dimensional F_2-vector space. The list of length 8 that is returned is
supposed to be a basis of this (or rather a set of representatives in
K^×^). But the S-2-Selmer group is a subgroup of K^×^ mod squares, so 121
is the zero vector and hence the output is not linearly independent. The
problem lies in the following:
{{{
sage: K._S_class_group_and_units(S)
([-7/13*a^2 - 140/13*a + 36/13,
14/13*a^2 + 267/13*a - 85/13,
7/13*a^2 + 127/13*a - 49/13,
-1,
1/13*a^2 + 20/13*a - 7/13,
1/13*a^2 - 19/13*a + 6/13],
[(Fractional ideal (11, a - 2), 2, 121),
(Fractional ideal (19, 1/13*a^2 - 45/13*a - 332/13),
2,
10/13*a^2 + 44/13*a - 4555/13)])
}}}
The 121 in there is supposed to be such that (11, a-2)^2^ = (121). But
(11, a-2)^2^ is *not* principal (in fact, (11, a-2) is a generator of the
cyclic subgroup of order 6). It is just in the subgroup of the class group
generated by the primes in S.
I'll shortly upload a patch that fixes this (partially suggested to me by
Zev Klagsbrun).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14489>
Sage <http://www.sagemath.org>
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