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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 25857.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25857.h1 | 25857s2 | \([1, -1, 1, -500, 3098]\) | \(8615125/2601\) | \(4165795413\) | \([2]\) | \(12288\) | \(0.55100\) | |
| 25857.h2 | 25857s1 | \([1, -1, 1, 85, 290]\) | \(42875/51\) | \(-81682263\) | \([2]\) | \(6144\) | \(0.20443\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25857.h have rank \(1\).
Complex multiplication
The elliptic curves in class 25857.h do not have complex multiplication.Modular form 25857.2.a.h
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
