Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 67760.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67760.k1 | 67760bo3 | \([0, -1, 0, -254261, -51537539]\) | \(-250523582464/13671875\) | \(-99207416000000000\) | \([]\) | \(583200\) | \(2.0196\) | |
67760.k2 | 67760bo1 | \([0, -1, 0, -2581, 56861]\) | \(-262144/35\) | \(-253970984960\) | \([]\) | \(64800\) | \(0.92094\) | \(\Gamma_0(N)\)-optimal |
67760.k3 | 67760bo2 | \([0, -1, 0, 16779, -148355]\) | \(71991296/42875\) | \(-311114456576000\) | \([]\) | \(194400\) | \(1.4703\) |
Rank
sage: E.rank()
The elliptic curves in class 67760.k have rank \(0\).
Complex multiplication
The elliptic curves in class 67760.k do not have complex multiplication.Modular form 67760.2.a.k
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.