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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 23232.bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 23232.bo1 | 23232cz6 | \([0, -1, 0, -186017, -30817983]\) | \(3065617154/9\) | \(2089818390528\) | \([2]\) | \(81920\) | \(1.5933\) | |
| 23232.bo2 | 23232cz4 | \([0, -1, 0, -31137, 2124993]\) | \(28756228/3\) | \(348303065088\) | \([2]\) | \(40960\) | \(1.2467\) | |
| 23232.bo3 | 23232cz3 | \([0, -1, 0, -11777, -465375]\) | \(1556068/81\) | \(9404182757376\) | \([2, 2]\) | \(40960\) | \(1.2467\) | |
| 23232.bo4 | 23232cz2 | \([0, -1, 0, -2097, 28305]\) | \(35152/9\) | \(261227298816\) | \([2, 2]\) | \(20480\) | \(0.90017\) | |
| 23232.bo5 | 23232cz1 | \([0, -1, 0, 323, 2653]\) | \(2048/3\) | \(-5442235392\) | \([2]\) | \(10240\) | \(0.55360\) | \(\Gamma_0(N)\)-optimal |
| 23232.bo6 | 23232cz5 | \([0, -1, 0, 7583, -1863167]\) | \(207646/6561\) | \(-1523477606694912\) | \([2]\) | \(81920\) | \(1.5933\) |
Rank
sage: E.rank()
The elliptic curves in class 23232.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 23232.bo do not have complex multiplication.Modular form 23232.2.a.bo
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
