Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 66654.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66654.c1 | 66654m2 | \([1, -1, 0, -128646, -81027054]\) | \(-2181825073/25039686\) | \(-2702236917245032566\) | \([]\) | \(1520640\) | \(2.2187\) | |
66654.c2 | 66654m1 | \([1, -1, 0, 14184, 2871288]\) | \(2924207/34776\) | \(-3752962039304856\) | \([]\) | \(506880\) | \(1.6693\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66654.c have rank \(1\).
Complex multiplication
The elliptic curves in class 66654.c do not have complex multiplication.Modular form 66654.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.