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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 12936.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12936.d1 | 12936t4 | \([0, -1, 0, -1863584, 979821228]\) | \(2970658109581346/2139291\) | \(515451795167232\) | \([2]\) | \(196608\) | \(2.1349\) | |
| 12936.d2 | 12936t3 | \([0, -1, 0, -268144, -31653236]\) | \(8849350367426/3314597517\) | \(798636202552387584\) | \([2]\) | \(196608\) | \(2.1349\) | |
| 12936.d3 | 12936t2 | \([0, -1, 0, -117224, 15131964]\) | \(1478729816932/38900169\) | \(4686403566265344\) | \([2, 2]\) | \(98304\) | \(1.7883\) | |
| 12936.d4 | 12936t1 | \([0, -1, 0, 1356, 760068]\) | \(9148592/8301447\) | \(-250024176154368\) | \([4]\) | \(49152\) | \(1.4417\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12936.d have rank \(2\).
Complex multiplication
The elliptic curves in class 12936.d do not have complex multiplication.Modular form 12936.2.a.d
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.
