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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 134640j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.dq4 | 134640j1 | \([0, 0, 0, -2696907, 1704694394]\) | \(726497538898787209/1038579300\) | \(3101181172531200\) | \([2]\) | \(2211840\) | \(2.2438\) | \(\Gamma_0(N)\)-optimal |
| 134640.dq3 | 134640j2 | \([0, 0, 0, -2721387, 1672170266]\) | \(746461053445307689/27443694341250\) | \(81946432203863040000\) | \([2]\) | \(4423680\) | \(2.5904\) | |
| 134640.dq2 | 134640j3 | \([0, 0, 0, -3433467, 700554026]\) | \(1499114720492202169/796539777000000\) | \(2378455029485568000000\) | \([2]\) | \(6635520\) | \(2.7931\) | |
| 134640.dq1 | 134640j4 | \([0, 0, 0, -31732347, -68275136086]\) | \(1183430669265454849849/10449720703125000\) | \(31202698824000000000000\) | \([2]\) | \(13271040\) | \(3.1397\) |
Rank
sage: E.rank()
The elliptic curves in class 134640j have rank \(2\).
Complex multiplication
The elliptic curves in class 134640j do not have complex multiplication.Modular form 134640.2.a.j
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
