Show commands:
SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 286110.ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286110.ex1 | 286110ex2 | \([1, -1, 1, -1870401263, 31018126174967]\) | \(41125104693338423360329/179205840000000000\) | \(3153357536259957840000000000\) | \([2]\) | \(230031360\) | \(4.1295\) | |
| 286110.ex2 | 286110ex1 | \([1, -1, 1, -59272943, 962813930231]\) | \(-1308796492121439049/22000592486400000\) | \(-387128757183212578406400000\) | \([2]\) | \(115015680\) | \(3.7829\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286110.ex have rank \(1\).
Complex multiplication
The elliptic curves in class 286110.ex do not have complex multiplication.Modular form 286110.2.a.ex
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
