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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 19350.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19350.e1 | 19350ba3 | \([1, -1, 0, -188667, -31280009]\) | \(65202655558249/512820150\) | \(5841342021093750\) | \([2]\) | \(147456\) | \(1.8535\) | |
| 19350.e2 | 19350ba2 | \([1, -1, 0, -19917, 276241]\) | \(76711450249/41602500\) | \(473878476562500\) | \([2, 2]\) | \(73728\) | \(1.5069\) | |
| 19350.e3 | 19350ba1 | \([1, -1, 0, -15417, 739741]\) | \(35578826569/51600\) | \(587756250000\) | \([2]\) | \(36864\) | \(1.1604\) | \(\Gamma_0(N)\)-optimal |
| 19350.e4 | 19350ba4 | \([1, -1, 0, 76833, 2114491]\) | \(4403686064471/2721093750\) | \(-30994958496093750\) | \([2]\) | \(147456\) | \(1.8535\) |
Rank
sage: E.rank()
The elliptic curves in class 19350.e have rank \(1\).
Complex multiplication
The elliptic curves in class 19350.e do not have complex multiplication.Modular form 19350.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
