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SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 283920hf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 283920.hf3 | 283920hf1 | \([0, 1, 0, -53460, 4739580]\) | \(13674725584/945\) | \(1167701633280\) | \([2]\) | \(884736\) | \(1.3692\) | \(\Gamma_0(N)\)-optimal |
| 283920.hf2 | 283920hf2 | \([0, 1, 0, -56840, 4102788]\) | \(4108974916/893025\) | \(4413912173798400\) | \([2, 2]\) | \(1769472\) | \(1.7158\) | |
| 283920.hf4 | 283920hf3 | \([0, 1, 0, 125680, 25202100]\) | \(22208984782/40516875\) | \(-400521660215040000\) | \([2]\) | \(3538944\) | \(2.0624\) | |
| 283920.hf1 | 283920hf4 | \([0, 1, 0, -293440, -57697132]\) | \(282678688658/18600435\) | \(183870969982801920\) | \([2]\) | \(3538944\) | \(2.0624\) |
Rank
sage: E.rank()
The elliptic curves in class 283920hf have rank \(1\).
Complex multiplication
The elliptic curves in class 283920hf do not have complex multiplication.Modular form 283920.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
