Opened 12 years ago

Closed 12 years ago

# descend_to method for elliptic curves

Reported by: Owned by: ebeyerstedt cremona minor sage-4.5.2 elliptic curves descend, subfield, isomorphic, elliptic curve adeines, cremona, jeremywest sage-4.5.2.alpha1 Erin Beyerstedt, Jeremy West Aly Deines N/A

### Description

Given a subfield K and an elliptic curve E defined over a field L, this function determines whether there exists an elliptic curve over K which is isomorphic over L to E. If one exists, it finds it.

### comment:1 Changed 12 years ago by ebeyerstedt

• Status changed from new to needs_review

This has been implemented in the patch. Please review.

### comment:2 Changed 12 years ago by ebeyerstedt

Note: This function does not yet work when j = 0, 1728.

### comment:3 Changed 12 years ago by ebeyerstedt

The update handles the cases for j=0,1728.

### comment:5 Changed 12 years ago by ebeyerstedt

• Cc cremona added

### comment:6 Changed 12 years ago by aly.deines

• Cc jeremywest added
• Status changed from needs_review to needs_work

The code for handling j=0,1728 needs to be cleaned up a little. Also, this function currently does not properly handle the following case

F.<b> = QuadraticField?(23) K.<a> = F.extension(x3+5) E = EllipticCurve?(j = 1728*b).change_ring(K) E.descend_to(F)

It returns none when it should descend to the subfield F.

### comment:7 Changed 12 years ago by cremona

It looks to me as though the curve returned is (sometimes) a twist of the original, rather than isomorphic -- but I have been flying all night so am not reliable!

You can check if there is an embedding of K in self.base_ring() like this:

```sage: X = polygen(QQ)
sage: K.<a> = NumberField(X^4 - X^3 + 2*X^2 + X + 1)
sage: QQ.embeddings(K)
[Ring Coercion morphism:
From: Rational Field
To:   Number Field in a with defining polynomial x^4 - x^3 + 2*x^2 + x + 1]
```

### comment:8 Changed 12 years ago by ebeyerstedt

• Status changed from needs_work to needs_review

This new patch should work in general. It uses the newly implemented preimage function for number field homomorphisms. Be sure to apply the patch from #9403 first.

### Changed 12 years ago by ebeyerstedt

Replaces previous patch.

### comment:9 Changed 12 years ago by aly.deines

• Status changed from needs_review to positive_review

### comment:10 Changed 12 years ago by ddrake

• Merged in set to sage-4.5.2.alpha1
• Resolution set to fixed
• Status changed from positive_review to closed

### comment:11 Changed 12 years ago by mpatel

• Milestone changed from sage-5.0 to sage-4.5.2

### comment:12 Changed 12 years ago by mpatel

• Reviewers set to Alyson Deines

I'm entering a guess for the Reviewer(s) field. Please correct me if I'm wrong.

### comment:13 Changed 12 years ago by aly.deines

• Reviewers changed from Alyson Deines to Aly Deines

### comment:14 Changed 8 years ago by cremona

See follow-up ticket at #16456 where it is explained why the implementation here is deficient and needs fixing.

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