Show commands:
SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 9450.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9450.cm1 | 9450dc2 | \([1, -1, 1, -4835, -3293]\) | \(25397889795/14680064\) | \(7223692492800\) | \([]\) | \(18144\) | \(1.1569\) | |
| 9450.cm2 | 9450dc1 | \([1, -1, 1, -3260, 72447]\) | \(5674764464955/43904\) | \(29635200\) | \([]\) | \(6048\) | \(0.60755\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9450.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 9450.cm do not have complex multiplication.Modular form 9450.2.a.cm
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
