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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1170b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1170.c3 | 1170b1 | \([1, -1, 0, -249, 1593]\) | \(-63378025803/812500\) | \(-21937500\) | \([6]\) | \(576\) | \(0.21793\) | \(\Gamma_0(N)\)-optimal |
| 1170.c2 | 1170b2 | \([1, -1, 0, -3999, 98343]\) | \(261984288445803/42250\) | \(1140750\) | \([6]\) | \(1152\) | \(0.56450\) | |
| 1170.c4 | 1170b3 | \([1, -1, 0, 876, 7568]\) | \(3774555693/3515200\) | \(-69189681600\) | \([2]\) | \(1728\) | \(0.76723\) | |
| 1170.c1 | 1170b4 | \([1, -1, 0, -4524, 71288]\) | \(520300455507/193072360\) | \(3800243261880\) | \([2]\) | \(3456\) | \(1.1138\) |
Rank
sage: E.rank()
The elliptic curves in class 1170b have rank \(1\).
Complex multiplication
The elliptic curves in class 1170b do not have complex multiplication.Modular form 1170.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
