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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2640.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2640.b1 | 2640o4 | \([0, -1, 0, -249996, 48194796]\) | \(6749703004355978704/5671875\) | \(1452000000\) | \([2]\) | \(6912\) | \(1.4936\) | |
| 2640.b2 | 2640o3 | \([0, -1, 0, -15621, 757296]\) | \(-26348629355659264/24169921875\) | \(-386718750000\) | \([2]\) | \(3456\) | \(1.1470\) | |
| 2640.b3 | 2640o2 | \([0, -1, 0, -3156, 63900]\) | \(13584145739344/1195803675\) | \(306125740800\) | \([2]\) | \(2304\) | \(0.94425\) | |
| 2640.b4 | 2640o1 | \([0, -1, 0, 219, 4500]\) | \(72268906496/606436875\) | \(-9702990000\) | \([2]\) | \(1152\) | \(0.59767\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2640.b have rank \(0\).
Complex multiplication
The elliptic curves in class 2640.b do not have complex multiplication.Modular form 2640.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
