EC question

[John Jones]

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

[Edgar Costa]

The isogeny classes are complete.
In particular, the isogeny class of the curve that you point is https://beta.lmfdb.org/EllipticCurve/2.2.5.1/2025.1/d/
and it has two curves linked by isogenies of degree 3.

[John Cremona]

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

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