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SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 58800hf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.x4 | 58800hf1 | \([0, -1, 0, -59208, -7523088]\) | \(-24389/12\) | \(-11294304000000000\) | \([2]\) | \(345600\) | \(1.7855\) | \(\Gamma_0(N)\)-optimal |
| 58800.x2 | 58800hf2 | \([0, -1, 0, -1039208, -407363088]\) | \(131872229/18\) | \(16941456000000000\) | \([2]\) | \(691200\) | \(2.1321\) | |
| 58800.x3 | 58800hf3 | \([0, -1, 0, -549208, 752956912]\) | \(-19465109/248832\) | \(-234198687744000000000\) | \([2]\) | \(1728000\) | \(2.5903\) | |
| 58800.x1 | 58800hf4 | \([0, -1, 0, -16229208, 25088316912]\) | \(502270291349/1889568\) | \(1778446285056000000000\) | \([2]\) | \(3456000\) | \(2.9368\) |
Rank
sage: E.rank()
The elliptic curves in class 58800hf have rank \(0\).
Complex multiplication
The elliptic curves in class 58800hf do not have complex multiplication.Modular form 58800.2.a.hf
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
