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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 32448cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32448.q3 | 32448cl1 | \([0, -1, 0, -26589, -1659555]\) | \(420616192/117\) | \(578290332672\) | \([2]\) | \(86016\) | \(1.2387\) | \(\Gamma_0(N)\)-optimal |
| 32448.q2 | 32448cl2 | \([0, -1, 0, -29969, -1207311]\) | \(37642192/13689\) | \(1082559502761984\) | \([2, 2]\) | \(172032\) | \(1.5853\) | |
| 32448.q4 | 32448cl3 | \([0, -1, 0, 91711, -8629791]\) | \(269676572/257049\) | \(-81312247096344576\) | \([2]\) | \(344064\) | \(1.9319\) | |
| 32448.q1 | 32448cl4 | \([0, -1, 0, -205729, 35104705]\) | \(3044193988/85293\) | \(26980713761144832\) | \([2]\) | \(344064\) | \(1.9319\) |
Rank
sage: E.rank()
The elliptic curves in class 32448cl have rank \(0\).
Complex multiplication
The elliptic curves in class 32448cl do not have complex multiplication.Modular form 32448.2.a.cl
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 Cremona 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 Cremona labels.
