Show commands:
SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 350658.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
350658.cp1 | 350658cp4 | \([1, -1, 1, -7505411, -7912085893]\) | \(36204575259448513/1527466248\) | \(1972673733020610312\) | \([2]\) | \(13271040\) | \(2.5906\) | |
350658.cp2 | 350658cp2 | \([1, -1, 1, -492251, -110646709]\) | \(10214075575873/1806590016\) | \(2333153138779197504\) | \([2, 2]\) | \(6635520\) | \(2.2440\) | |
350658.cp3 | 350658cp1 | \([1, -1, 1, -143771, 19406027]\) | \(254478514753/21762048\) | \(28104987931840512\) | \([4]\) | \(3317760\) | \(1.8974\) | \(\Gamma_0(N)\)-optimal |
350658.cp4 | 350658cp3 | \([1, -1, 1, 945229, -636764389]\) | \(72318867421247/177381135624\) | \(-229082054963240827656\) | \([2]\) | \(13271040\) | \(2.5906\) |
Rank
sage: E.rank()
The elliptic curves in class 350658.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 350658.cp do not have complex multiplication.Modular form 350658.2.a.cp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.