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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 134640.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.f1 | 134640eq4 | \([0, 0, 0, -790563, -270552638]\) | \(36599544910739522/104135625\) | \(155473655040000\) | \([2]\) | \(1441792\) | \(1.9542\) | |
| 134640.f2 | 134640eq3 | \([0, 0, 0, -141843, 15292978]\) | \(211392685378082/54661514655\) | \(81609204087797760\) | \([2]\) | \(1441792\) | \(1.9542\) | |
| 134640.f3 | 134640eq2 | \([0, 0, 0, -50043, -4113542]\) | \(18566337396964/952031025\) | \(710687352038400\) | \([2, 2]\) | \(720896\) | \(1.6077\) | |
| 134640.f4 | 134640eq1 | \([0, 0, 0, 1977, -253658]\) | \(4579058864/151590615\) | \(-28290446933760\) | \([2]\) | \(360448\) | \(1.2611\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 134640.f have rank \(1\).
Complex multiplication
The elliptic curves in class 134640.f do not have complex multiplication.Modular form 134640.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
