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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