Show commands:
SageMath
E = EllipticCurve("gw1")
E.isogeny_class()
Elliptic curves in class 162240.gw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162240.gw1 | 162240et2 | \([0, 1, 0, -7159065, -7361775225]\) | \(2052450196928704/4317958125\) | \(85368664634872320000\) | \([2]\) | \(6193152\) | \(2.7095\) | |
162240.gw2 | 162240et1 | \([0, 1, 0, -293440, -195435850]\) | \(-9045718037056/48125390625\) | \(-14866692390225000000\) | \([2]\) | \(3096576\) | \(2.3630\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162240.gw have rank \(1\).
Complex multiplication
The elliptic curves in class 162240.gw do not have complex multiplication.Modular form 162240.2.a.gw
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.