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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1638.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1638.e1 | 1638c3 | \([1, -1, 0, -1932, -14896]\) | \(40530337875/18264064\) | \(359491571712\) | \([2]\) | \(1728\) | \(0.91259\) | |
| 1638.e2 | 1638c1 | \([1, -1, 0, -957, 11637]\) | \(3592121380875/246064\) | \(6643728\) | \([6]\) | \(576\) | \(0.36329\) | \(\Gamma_0(N)\)-optimal |
| 1638.e3 | 1638c2 | \([1, -1, 0, -897, 13113]\) | \(-2958077788875/946054564\) | \(-25543473228\) | \([6]\) | \(1152\) | \(0.70986\) | |
| 1638.e4 | 1638c4 | \([1, -1, 0, 6708, -116848]\) | \(1695802078125/1272491584\) | \(-25046451847872\) | \([2]\) | \(3456\) | \(1.2592\) |
Rank
sage: E.rank()
The elliptic curves in class 1638.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1638.e do not have complex multiplication.Modular form 1638.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
