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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 13680.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13680.a1 | 13680bh4 | \([0, 0, 0, -437763, 111482498]\) | \(3107086841064961/570\) | \(1702010880\) | \([2]\) | \(73728\) | \(1.6070\) | |
| 13680.a2 | 13680bh3 | \([0, 0, 0, -31683, 1154882]\) | \(1177918188481/488703750\) | \(1459261578240000\) | \([2]\) | \(73728\) | \(1.6070\) | |
| 13680.a3 | 13680bh2 | \([0, 0, 0, -27363, 1741538]\) | \(758800078561/324900\) | \(970146201600\) | \([2, 2]\) | \(36864\) | \(1.2604\) | |
| 13680.a4 | 13680bh1 | \([0, 0, 0, -1443, 36002]\) | \(-111284641/123120\) | \(-367634350080\) | \([2]\) | \(18432\) | \(0.91382\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 13680.a have rank \(2\).
Complex multiplication
The elliptic curves in class 13680.a do not have complex multiplication.Modular form 13680.2.a.a
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.
