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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 29760bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29760.cs3 | 29760bd1 | \([0, 1, 0, -6945, -200385]\) | \(141339344329/17141760\) | \(4493609533440\) | \([2]\) | \(55296\) | \(1.1583\) | \(\Gamma_0(N)\)-optimal |
| 29760.cs2 | 29760bd2 | \([0, 1, 0, -27425, 1532223]\) | \(8702409880009/1120910400\) | \(293839935897600\) | \([2, 2]\) | \(110592\) | \(1.5049\) | |
| 29760.cs4 | 29760bd3 | \([0, 1, 0, 41695, 8070975]\) | \(30579142915511/124675335000\) | \(-32682891018240000\) | \([2]\) | \(221184\) | \(1.8514\) | |
| 29760.cs1 | 29760bd4 | \([0, 1, 0, -424225, 106208063]\) | \(32208729120020809/658986840\) | \(172749446184960\) | \([4]\) | \(221184\) | \(1.8514\) |
Rank
sage: E.rank()
The elliptic curves in class 29760bd have rank \(1\).
Complex multiplication
The elliptic curves in class 29760bd do not have complex multiplication.Modular form 29760.2.a.bd
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.
