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SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 75690.bs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75690.bs1 | 75690z4 | \([1, -1, 1, -966467, -365453909]\) | \(8527173507/200\) | \(2341581485448600\) | \([2]\) | \(1161216\) | \(2.0612\) | |
| 75690.bs2 | 75690z3 | \([1, -1, 1, -58187, -6138341]\) | \(-1860867/320\) | \(-3746530376717760\) | \([2]\) | \(580608\) | \(1.7146\) | |
| 75690.bs3 | 75690z2 | \([1, -1, 1, -20342, 296991]\) | \(57960603/31250\) | \(501882177093750\) | \([2]\) | \(387072\) | \(1.5119\) | |
| 75690.bs4 | 75690z1 | \([1, -1, 1, 4888, 34599]\) | \(804357/500\) | \(-8030114833500\) | \([2]\) | \(193536\) | \(1.1653\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75690.bs have rank \(1\).
Complex multiplication
The elliptic curves in class 75690.bs do not have complex multiplication.Modular form 75690.2.a.bs
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
