#13376: Optional SPKG for SmallJac
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Reporter: pavpanchekha | Owner: tbd
Type: enhancement | Status: new
Priority: major | Milestone:
Component: optional packages | Keywords:
Work issues: | Report Upstream: N/A
Reviewers: | Authors:
Merged in: | Dependencies:
Stopgaps: |
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Smalljac is a library and set of programs for computing L-polynomials and
Jacobian group structures of low genus curves over finite fields. Given a
curve C of genus 1 or 2 defined over Q or a quadratic number field, it
will efficiently compute either the sequence of L-polynomials or Jacobian
group structures arising from the reduction of C at all primes of good
reduction up to a specified norm bound.
This set of SPKGs and the corresponding patch to Sage 5.2 add support for
using smallJac from Sage.
* `EllipticCurve.aplist` can be run with `algorithm="smalljac"`, yielding
run times of six times faster on a single core.
* `EllipticCurve.aplists` provides more usable representation the above,
and supports number fields (multiple reductions per prime norm). See the
documentation for more, but in general `aplist` couldn't be made very easy
to use for the number field case.
* `EllipticCurve.grouplists` gives the abelian groups isomorphic to
various reductions of a curve.
* `EllipticCurve.lpolylists` yields the L-Polynomials of reductions of a
curve.
* `EllipticCurve.guess_sato_tate_group` attempts to guess the Sato-Tate
group of the Jacobian of a curve of genus 1 or 2 based on the distribution
of its L-polynomial coefficients.
* `EllipticCurve.moments` yields the moments of the distribution of
coefficients of the L-Polynomials.
* `EllipticCurve.trace_histogram` gives a very simple histogram for the
distribution of traces of Frobenius of a curve, reduced at a range of
primes.
For all of the above methods (except `aplist`), identical functionality
exists (with the same name) for genus 2 curves. All of the above methods
also parallelize over multiple cores (currently, to a power of two cores).
For large problems, this achieves approximately linear speedup; for
example, computing the first billion traces of Frobenius of an ordinary
elliptic curve is 178 times faster on 32 cores.
For each of these features, there is Sage documentation docstring-style
documentation.
The SPKGs only compile and work on 64-bit machines, since SmallJac only
supports 64-bit architectures, and require GMP. Two SPKGs are provided,
since SmallJac depends on ff_poly, but so does some of Drew Sutherland's
other code, so it makes sense to split the two. They've been tested to
work on a completely clean install of Sage on at least two differently-
configured machines.
The patch has been tested to apply cleanly to 5.2.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13376>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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