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SageMath
sage: E = EllipticCurve("265200.bo1")
sage: E.isogeny_class()
Elliptic curves in class 265200.bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 265200.bo1 | 265200bo5 | [0, -1, 0, -8069608, 8824545712] | [2] | 8388608 | |
| 265200.bo2 | 265200bo3 | [0, -1, 0, -555608, 108305712] | [2, 2] | 4194304 | |
| 265200.bo3 | 265200bo2 | [0, -1, 0, -217608, -37710288] | [2, 2] | 2097152 | |
| 265200.bo4 | 265200bo1 | [0, -1, 0, -215608, -38462288] | [2] | 1048576 | \(\Gamma_0(N)\)-optimal |
| 265200.bo5 | 265200bo4 | [0, -1, 0, 88392, -135630288] | [2] | 4194304 | |
| 265200.bo6 | 265200bo6 | [0, -1, 0, 1550392, 731681712] | [2] | 8388608 |
Rank
sage: E.rank()
The elliptic curves in class 265200.bo have rank \(0\).
Modular form 265200.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
