Bug in elliptic curve isogeny

[cremona]

There is a bug in the code to compute 5-isogenies of elliptic curves of j-invariant 1728, when 5 is a square.

sage: K.<a> = NumberField(x^4 - 5*x^2 + 5)
sage: E = EllipticCurve([a^2 + a + 1, a^3 + a^2 + a + 1, a^2 + a,
17*a^3 + 34*a^2 - 16*a - 37, 54*a^3 + 105*a^2 - 66*a - 135])
sage: E.j_invariant()
1728
sage: K(5).is_square()
True
sage: E.isogenies_prime_degree(5)
ValueError: The polynomial does not define a finite subgroup of the elliptic curve.

or more directly

sage: from sage.schemes.elliptic_curves.isogeny_small_degree import isogenies_5_1728
sage: isogenies_5_1728(E)
ValueError: The polynomial does not define a finite subgroup of the elliptic curve.

I wrote this code about 5 years ago, and will fix it.

[cremona]

I found the bug: on line 868, a*(beta**2-2)/6 should be (beta**2-2*a)/6.

The bug was not caught by the doctests since the only example where this code was tested had a=1.

[cremona]

To review, run the code in the description before and after; note that a doctest is added with this example.

---

New commits:

​a88c9a5

#19840: isogeny bug fix and test

[cremona]

There is a small merge conflict with #19689 so I will rebase this on that and make that ticket a dependency.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ff0ba0c	#19689: unit scaling for Weierstrass models of elliptic curves over number fields
​80595c1	#19840: isogeny bug fix and test

[chapoton]

two details:

• The following is not formatted correctly

  See `trac``19840`:
  

but should be

See :trac:`19840`:

• A typo in the first line: "with repect"

[cremona]

Replying to chapoton:

two details:

• The following is not formatted correctly

  See `trac``19840`:
  

but should be

See :trac:`19840`:

• A typo in the first line: "with repect"

Thanks, I am fixing those now and hope we can this into 7.0.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e6ed40d

#19840: correct typos after review

[cremona]

Don't forget to fill in the "reviewer" box :)

[chapoton]

sorry, still not formatted correctly, should be in fact

See :trac:`19840`::

with a double colon at the end.

Once done, you can set a positive review.

[cremona]

Sorry, will do. It is so hard to actually test correct formatting of docstrings since it takes a very long time to build doc and the output is thousamds of lines somewhere in which is the relevant line or two.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8ce0e65

#19840 last typo

[cremona]

7.0 please!