Show commands:
SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 41616.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41616.be1 | 41616ca4 | \([0, 0, 0, -4703475, -2713145294]\) | \(159661140625/48275138\) | \(3479401356824541339648\) | \([2]\) | \(1990656\) | \(2.8385\) | |
41616.be2 | 41616ca3 | \([0, 0, 0, -4287315, -3416372462]\) | \(120920208625/19652\) | \(1416406007256072192\) | \([2]\) | \(995328\) | \(2.4920\) | |
41616.be3 | 41616ca2 | \([0, 0, 0, -1790355, 921845842]\) | \(8805624625/2312\) | \(166636000853655552\) | \([2]\) | \(663552\) | \(2.2892\) | |
41616.be4 | 41616ca1 | \([0, 0, 0, -125715, 10621906]\) | \(3048625/1088\) | \(78416941578190848\) | \([2]\) | \(331776\) | \(1.9427\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 41616.be have rank \(1\).
Complex multiplication
The elliptic curves in class 41616.be do not have complex multiplication.Modular form 41616.2.a.be
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.