Show commands:
SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 489762s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.s1 | 489762s1 | \([1, -1, 0, -87333, 9943649]\) | \(1703492778807/2384732\) | \(103124279863332\) | \([2]\) | \(2598912\) | \(1.5926\) | \(\Gamma_0(N)\)-optimal |
489762.s2 | 489762s2 | \([1, -1, 0, -62763, 15638975]\) | \(-632291491287/2072502446\) | \(-89622365221225746\) | \([2]\) | \(5197824\) | \(1.9392\) |
Rank
sage: E.rank()
The elliptic curves in class 489762s have rank \(2\).
Complex multiplication
The elliptic curves in class 489762s do not have complex multiplication.Modular form 489762.2.a.s
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.