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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 81312.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81312.f1 | 81312e4 | \([0, -1, 0, -27144, -1712280]\) | \(2438569736/21\) | \(19047823872\) | \([2]\) | \(163840\) | \(1.1403\) | |
| 81312.f2 | 81312e3 | \([0, -1, 0, -5969, 150273]\) | \(3241792/567\) | \(4114329956352\) | \([2]\) | \(163840\) | \(1.1403\) | |
| 81312.f3 | 81312e1 | \([0, -1, 0, -1734, -25056]\) | \(5088448/441\) | \(50000537664\) | \([2, 2]\) | \(81920\) | \(0.79374\) | \(\Gamma_0(N)\)-optimal |
| 81312.f4 | 81312e2 | \([0, -1, 0, 1896, -119436]\) | \(830584/7203\) | \(-6533403588096\) | \([2]\) | \(163840\) | \(1.1403\) |
Rank
sage: E.rank()
The elliptic curves in class 81312.f have rank \(0\).
Complex multiplication
The elliptic curves in class 81312.f do not have complex multiplication.Modular form 81312.2.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.
