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
On Tue, 30 Apr 2024 at 17:08, John Cremona <john.crem...@gmail.com> wrote:
> 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.crem...@gmail.com>;
>> *Date: * Tue, Apr 30, 2024 11:35 PM
>> *To: * "John Jones"<j...@asu.edu>;
>> *Cc: * "lmfdb-support"<lmfdb-supp...@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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>>
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