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SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 78400.hf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78400.hf1 | 78400hn3 | \([0, 1, 0, -643533, -208075687]\) | \(-250523582464/13671875\) | \(-1608482421875000000\) | \([]\) | \(995328\) | \(2.2517\) | |
| 78400.hf2 | 78400hn1 | \([0, 1, 0, -6533, 223313]\) | \(-262144/35\) | \(-4117715000000\) | \([]\) | \(110592\) | \(1.1531\) | \(\Gamma_0(N)\)-optimal |
| 78400.hf3 | 78400hn2 | \([0, 1, 0, 42467, -560687]\) | \(71991296/42875\) | \(-5044200875000000\) | \([]\) | \(331776\) | \(1.7024\) |
Rank
sage: E.rank()
The elliptic curves in class 78400.hf have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.hf do not have complex multiplication.Modular form 78400.2.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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
