#16779: Isogeny construction fails over relative number fields
-------------------------+-------------------------------------------------
Reporter: cremona | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.3
Component: elliptic | Keywords: isogeny relative number field
curves | Authors: John Cremona
Merged in: | Report Upstream: N/A
Reviewers: | Branch:
Work issues: | Dependencies:
Commit: |
Stopgaps: |
-------------------------+-------------------------------------------------
In 6.3.beta8:
{{{
sage: pol26 = hilbert_class_polynomial(-4*26)
sage: pol =
NumberField(pol26,'a').optimized_representation()[0].polynomial()
sage: K.<a> = NumberField(pol)
sage: j = pol26.roots(K)[0][0]
sage: E = EllipticCurve(j=j)
sage: L.<b> = K.extension(x^2+26)
sage: EL = E.change_ring(L)
sage: EL.isogenies_prime_degree(2)
<boom>
AttributeError: 'MPolynomial_polydict' object has no attribute 'gcd'
}}}
The problem is that the isogeny construction code uses 2-variable
polynomial rings where univariate polynomials would suffice. This can be
fixed by using pol.univariate_polynomial() instead of pol in a few places:
possibly not the best solution, but it does work. After the changes to be
posted:
{{{
sage: EL.isogenies_prime_degree(2)
[Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 +
(-1732240226222259558661029888*a^5-188333428554736651445698560*a^4-3673289955722628245763686400*a^3-5804899323109453402219118592*a^2-2971541838129936761551454208*a-1562374967437103565141073920)*x
+
(-94728391892057339794161932691112485453824*a^5-10299104336936650483469675988569712230400*a^4-200875632138916380192904691609307264843776*a^3-317443718053271064319198841938037520203776*a^2-162500198012313692944394229567905247264768*a-85439228322553980844209253657171793543168)
over Number Field in b with defining polynomial x^2 + 26 over its base
field to Elliptic Curve defined by y^2 = x^3 +
(-1732286128907119084224380928*a^5-188338419208778539462164480*a^4-3673387294340771630546288640*a^3-5805053147310602916403740672*a^2-2971620581105535693881868288*a-1562416368858421516757729280)*x
+
(-94723120436085499486994076285991739457536*a^5-10298531211245638927893552385996911280128*a^4-200864453789643413434655770571076062412800*a^3-317426052910331005519875176884304663805952*a^2-162491155183420182869799603998938865074176*a-85434473790978235642424626625064973893632)
over Number Field in b with defining polynomial x^2 + 26 over its base
field]
}}}
Note, however, that {{{EL.isogenies_prime_degree(3)}}}, while it works
correctly and finds two 3-isogenies, does give some warnings:
{{{
sage: iso = EL.isogenies_prime_degree(3); len(iso)
verbose 0 (3525: multi_polynomial_ideal.py, groebner_basis) Warning:
falling back to very slow toy implementation.
verbose 0 (3525: multi_polynomial_ideal.py, groebner_basis) Warning:
falling back to very slow toy implementation.
2
}}}
--
Ticket URL: <http://trac.sagemath.org/ticket/16779>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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