Show commands:
SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 257600.ez
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
257600.ez1 | 257600ez2 | \([0, -1, 0, -37633, -1204863]\) | \(5756278756/2705927\) | \(2770869248000000\) | \([2]\) | \(1228800\) | \(1.6574\) | |
257600.ez2 | 257600ez1 | \([0, -1, 0, 8367, -146863]\) | \(253012016/181447\) | \(-46450432000000\) | \([2]\) | \(614400\) | \(1.3109\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 257600.ez have rank \(0\).
Complex multiplication
The elliptic curves in class 257600.ez do not have complex multiplication.Modular form 257600.2.a.ez
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.