Opened 7 years ago

Closed 7 years ago

# Isogeny construction fails over relative number fields

Reported by: Owned by: cremona major sage-duplicate/invalid/wontfix elliptic curves isogeny relative number field John Cremona Chris Wuthrich N/A

### Description

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
```

### comment:1 Changed 7 years ago by cremona

See also #11327 where a more thorough solution was proposed, but not finished, in 2011. The patch (branch) posted there on 2014-08-09 solves both the issue reported there (constructing duals of 2-isogenies over function fields) and this one.

This ticket is therefore redundant and can be closed as a duplicate while #11327 is ready for review.

Last edited 7 years ago by cremona (previous) (diff)

### comment:2 Changed 7 years ago by cremona

• Milestone changed from sage-6.3 to sage-duplicate/invalid/wontfix
• Status changed from new to needs_review

### comment:3 Changed 7 years ago by wuthrich

• Reviewers set to Chris Wuthrich
• Status changed from needs_review to positive_review

I confirm that #11327 solves this ticket, too. This should be closed as won't fix.

### comment:4 Changed 7 years ago by vbraun

• Resolution set to duplicate
• Status changed from positive_review to closed
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