Show commands:
SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 51714.v
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51714.v1 | 51714s4 | \([1, -1, 1, -171905, 19000271]\) | \(159661140625/48275138\) | \(169867840648914018\) | \([2]\) | \(663552\) | \(2.0113\) | |
| 51714.v2 | 51714s3 | \([1, -1, 1, -156695, 23910059]\) | \(120920208625/19652\) | \(69150352391172\) | \([2]\) | \(331776\) | \(1.6647\) | |
| 51714.v3 | 51714s2 | \([1, -1, 1, -65435, -6424765]\) | \(8805624625/2312\) | \(8135335575432\) | \([2]\) | \(221184\) | \(1.4620\) | |
| 51714.v4 | 51714s1 | \([1, -1, 1, -4595, -73069]\) | \(3048625/1088\) | \(3828393211968\) | \([2]\) | \(110592\) | \(1.1154\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51714.v have rank \(0\).
Complex multiplication
The elliptic curves in class 51714.v do not have complex multiplication.Modular form 51714.2.a.v
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.
