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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 31680.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.c1 | 31680r4 | \([0, 0, 0, -56028, 2772848]\) | \(1628514404944/664335375\) | \(7934779201536000\) | \([2]\) | \(221184\) | \(1.7485\) | |
| 31680.c2 | 31680r2 | \([0, 0, 0, -25788, -1593808]\) | \(158792223184/16335\) | \(195104194560\) | \([2]\) | \(73728\) | \(1.1992\) | |
| 31680.c3 | 31680r1 | \([0, 0, 0, -1488, -28888]\) | \(-488095744/200475\) | \(-149653785600\) | \([2]\) | \(36864\) | \(0.85261\) | \(\Gamma_0(N)\)-optimal |
| 31680.c4 | 31680r3 | \([0, 0, 0, 11472, 315848]\) | \(223673040896/187171875\) | \(-139723056000000\) | \([2]\) | \(110592\) | \(1.4019\) |
Rank
sage: E.rank()
The elliptic curves in class 31680.c have rank \(0\).
Complex multiplication
The elliptic curves in class 31680.c do not have complex multiplication.Modular form 31680.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
