Show commands:
SageMath
E = EllipticCurve("gr1")
E.isogeny_class()
Elliptic curves in class 458640gr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.gr2 | 458640gr1 | \([0, 0, 0, 28077, 5026322]\) | \(6967871/35100\) | \(-12330560909721600\) | \([2]\) | \(3317760\) | \(1.7698\) | \(\Gamma_0(N)\)-optimal* |
458640.gr1 | 458640gr2 | \([0, 0, 0, -324723, 63802802]\) | \(10779215329/1232010\) | \(432802687931228160\) | \([2]\) | \(6635520\) | \(2.1164\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 458640gr have rank \(1\).
Complex multiplication
The elliptic curves in class 458640gr do not have complex multiplication.Modular form 458640.2.a.gr
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 Cremona 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 Cremona labels.