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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 42237.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42237.c1 | 42237b4 | \([1, -1, 0, -225873, -41261076]\) | \(37159393753/1053\) | \(36114158953197\) | \([2]\) | \(230400\) | \(1.7032\) | |
| 42237.c2 | 42237b3 | \([1, -1, 0, -63423, 5583006]\) | \(822656953/85683\) | \(2938622489636067\) | \([2]\) | \(230400\) | \(1.7032\) | |
| 42237.c3 | 42237b2 | \([1, -1, 0, -14688, -586845]\) | \(10218313/1521\) | \(52164896265729\) | \([2, 2]\) | \(115200\) | \(1.3566\) | |
| 42237.c4 | 42237b1 | \([1, -1, 0, 1557, -50760]\) | \(12167/39\) | \(-1337561442711\) | \([2]\) | \(57600\) | \(1.0100\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42237.c have rank \(1\).
Complex multiplication
The elliptic curves in class 42237.c do not have complex multiplication.Modular form 42237.2.a.c
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.
