Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 489762.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.a1 | 489762a2 | \([1, -1, 0, -110304, 3578134]\) | \(42180533641/22862322\) | \(80446652899473042\) | \([2]\) | \(7077888\) | \(1.9348\) | \(\Gamma_0(N)\)-optimal* |
489762.a2 | 489762a1 | \([1, -1, 0, 26586, 429664]\) | \(590589719/365148\) | \(-1284862246841628\) | \([2]\) | \(3538944\) | \(1.5882\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762.a have rank \(2\).
Complex multiplication
The elliptic curves in class 489762.a do not have complex multiplication.Modular form 489762.2.a.a
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.