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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 67240e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67240.g3 | 67240e1 | \([0, 0, 0, -3362, 68921]\) | \(55296/5\) | \(380008339280\) | \([2]\) | \(69120\) | \(0.96142\) | \(\Gamma_0(N)\)-optimal |
| 67240.g2 | 67240e2 | \([0, 0, 0, -11767, -413526]\) | \(148176/25\) | \(30400667142400\) | \([2, 2]\) | \(138240\) | \(1.3080\) | |
| 67240.g4 | 67240e3 | \([0, 0, 0, 21853, -2343314]\) | \(237276/625\) | \(-3040066714240000\) | \([2]\) | \(276480\) | \(1.6546\) | |
| 67240.g1 | 67240e4 | \([0, 0, 0, -179867, -29360346]\) | \(132304644/5\) | \(24320533713920\) | \([2]\) | \(276480\) | \(1.6546\) |
Rank
sage: E.rank()
The elliptic curves in class 67240e have rank \(0\).
Complex multiplication
The elliptic curves in class 67240e do not have complex multiplication.Modular form 67240.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
