Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 6864.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6864.x1 | 6864z2 | \([0, 1, 0, -668, -2376]\) | \(128962402000/66806883\) | \(17102562048\) | \([2]\) | \(4608\) | \(0.65554\) | |
6864.x2 | 6864z1 | \([0, 1, 0, -533, -4914]\) | \(1048576000000/1146717\) | \(18347472\) | \([2]\) | \(2304\) | \(0.30897\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6864.x have rank \(1\).
Complex multiplication
The elliptic curves in class 6864.x do not have complex multiplication.Modular form 6864.2.a.x
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.