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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 32448.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32448.cp1 | 32448cz2 | \([0, 1, 0, -66473, -2917641]\) | \(1643032000/767637\) | \(15176651490643968\) | \([2]\) | \(215040\) | \(1.7993\) | |
| 32448.cp2 | 32448cz1 | \([0, 1, 0, -55488, -5046534]\) | \(61162984000/41067\) | \(12686244172992\) | \([2]\) | \(107520\) | \(1.4527\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32448.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 32448.cp do not have complex multiplication.Modular form 32448.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
