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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 17640ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17640.v4 | 17640ca1 | \([0, 0, 0, 1617, -11662]\) | \(21296/15\) | \(-329341904640\) | \([2]\) | \(18432\) | \(0.89852\) | \(\Gamma_0(N)\)-optimal |
| 17640.v3 | 17640ca2 | \([0, 0, 0, -7203, -98098]\) | \(470596/225\) | \(19760514278400\) | \([2, 2]\) | \(36864\) | \(1.2451\) | |
| 17640.v1 | 17640ca3 | \([0, 0, 0, -95403, -11334778]\) | \(546718898/405\) | \(71137851402240\) | \([2]\) | \(73728\) | \(1.5917\) | |
| 17640.v2 | 17640ca4 | \([0, 0, 0, -60123, 5606678]\) | \(136835858/1875\) | \(329341904640000\) | \([2]\) | \(73728\) | \(1.5917\) |
Rank
sage: E.rank()
The elliptic curves in class 17640ca have rank \(0\).
Complex multiplication
The elliptic curves in class 17640ca do not have complex multiplication.Modular form 17640.2.a.ca
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.
