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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 37485.bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37485.bo1 | 37485br6 | \([1, -1, 0, -31637349, -68352594332]\) | \(40832710302042509761/91556816413125\) | \(7852472994862864738125\) | \([2]\) | \(3145728\) | \(3.0835\) | |
| 37485.bo2 | 37485br4 | \([1, -1, 0, -2696724, -220574957]\) | \(25288177725059761/14387797265625\) | \(1233985561207062890625\) | \([2, 2]\) | \(1572864\) | \(2.7369\) | |
| 37485.bo3 | 37485br2 | \([1, -1, 0, -1724319, 867935200]\) | \(6610905152742241/35128130625\) | \(3012803501687555625\) | \([2, 2]\) | \(786432\) | \(2.3903\) | |
| 37485.bo4 | 37485br1 | \([1, -1, 0, -1722114, 870273823]\) | \(6585576176607121/187425\) | \(16074715228425\) | \([2]\) | \(393216\) | \(2.0438\) | \(\Gamma_0(N)\)-optimal |
| 37485.bo5 | 37485br3 | \([1, -1, 0, -787194, 1806747025]\) | \(-629004249876241/16074715228425\) | \(-1378665971321641189425\) | \([2]\) | \(1572864\) | \(2.7369\) | |
| 37485.bo6 | 37485br5 | \([1, -1, 0, 10685421, -1764874490]\) | \(1573196002879828319/926055908203125\) | \(-79424223075714111328125\) | \([4]\) | \(3145728\) | \(3.0835\) |
Rank
sage: E.rank()
The elliptic curves in class 37485.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 37485.bo do not have complex multiplication.Modular form 37485.2.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.
