Re: EC question

[John Cremona]

I can confirm that your curve is isogenous (and not isomorphic) to the ones in the LMFDB.  The isogeny class computed by Sage from your curve has 6 curves in it.  That means that there is a bug in Sage's isogeny class code -- which I wrote most of.  I hope that it is something specific to j-invariant 0, which is (as always) treated separately.

I will investigate the Sage bug, and when it is fixed I will recompute all the isogeny classes in the LMFDB.  This will not be done very soon.

Thanks for the report!

John

On Tue, 30 Apr 2024 at 16:42, Zhengyu Tao <tao...@smail.nju.edu.cn> wrote:

Thanks for your reply! The coefficients of my curve is [0, -27*t^2, 0, 216*t^3*(t - 27), -432*t^4*(t - 27)^2] with t = -4320 - 1944\sqrt{5}.

 

------------------ Original ------------------

From:  "John Cremona"<john.c...@gmail.com>;

Date:  Tue, Apr 30, 2024 11:35 PM

To:  "John Jones"<j...@asu.edu>;

Cc:  "lmfdb-support"<lmfdb-...@googlegroups.com>; "taozhy"<tao...@smail.nju.edu.cn>;

Subject:  Re: EC question

 

The isogeny classes in the LMFDB are supposed to be complete.  I see that curve 2.2.5.1-2025.1-d2 has coefficients (0, 0, 1, 0, -34) while the isogenous curve d1 has coefficients (0, 0, 1, 0, 1), both with j-invariant 0 and conductor (45) over this field.   They are quadratic twists of each other by -3.

What are the coefficients of the curve you have?  If it is not isomorphic to either of these then there is a bug in Sage, which was used to compute the isogeny classes.

John Cremona

On Tue, 30 Apr 2024 at 16:08, John Jones <j...@asu.edu> wrote:

From the feedback page:

Hi LMFDB devs,

In a recent problem I'm working on, I need to compute a (CM) elliptic curve over Q(\sqrt{5}). When I searched it in LMFDB, it seems that it is not included. However, I found that my curve seems isogenous (over Q(\sqrt{5})) to the curve 2.2.5.1-2025.1-d2. In fact, I have constructed the isogeny from my curve to 2.2.5.1-2025.1-d2 using velu'formula.

My qusetion is: is the isogeny classes in LMFDB complete? I.e., is each isomorphism class (over the base field) in a isogeny class has a representative in LMFDB's "isogeny class"?

Best regards,
Zhengyu Tao

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[John Cremona]

This looks like the same bug as I reported at https://github.com/sagemath/sage/issues/36780 five months ago and supposedly fixed via a PR (https://github.com/sagemath/sage/pull/36786).  It's the same bug (over Q(sqrt(5)), j=0, missing a 5-isogeny) so clearly my fix was incorrect.

The curve is defined by 

sage: K.<r> = NumberField(x^2-5)
sage: t = -4320 - 1944*r
sage: E = EllipticCurve([0, -27*t^2, 0, 216*t^3*(t - 27), -432*t^4*(t - 27)^2])
sage: E.has_cm()
True
sage: E.cm_discriminant()
-75

sage: C = E.isogeny_class()

sage: len(C)
6

sage: C.matrix()[0]
(1, 25, 75, 3, 5, 15)

This is all correct, the class has size 6 and 3- and 5- isogenies suffice to fill it.  But the class contains two curves defined over Q for example

sage: E1 = C[5]; E1.ainvs()
(0, 0, 1, 0, 1)
sage: E1.j_invariant()
0

sage: E1.base_field() == K
True

and computing the isogeny class starting with E1 does not find the whole class as it misses 5-isogenies:

sage: len(E1.isogeny_class())
2

The problem is in the function possible_isogeny_degrees_cm():

sage: from sage.schemes.elliptic_curves.isogeny_class import isogeny_degrees_cm
sage: isogeny_degrees_cm(E1, verbose=True)
CM case, discriminant = -3
initial primes: {2, 3}
ramified primes: {3}
downward split primes: {}
downward inert primes: {}
Complete set of primes: {2, 3}
[2, 3]

Here, "downward" primes are sgrees of isogenies to curves with a strictly smaller endomorphism ring and in this case should inlude 5.  I cannot right now see the error in the code but am building the current development branch and will sort this out.

John

[John Cremona]

False alarm -- the current version of Safe computes this correctly, presumably because of the bugfix I made.

So this is no longer a Sage issue, but the LMFDB will need to be corrected.

John