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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 32448.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32448.be1 | 32448ch4 | \([0, -1, 0, -751937, 251213793]\) | \(37159393753/1053\) | \(1332380926476288\) | \([2]\) | \(344064\) | \(2.0038\) | |
32448.be2 | 32448ch3 | \([0, -1, 0, -211137, -33744543]\) | \(822656953/85683\) | \(108416329461792768\) | \([2]\) | \(344064\) | \(2.0038\) | |
32448.be3 | 32448ch2 | \([0, -1, 0, -48897, 3603105]\) | \(10218313/1521\) | \(1924550227132416\) | \([2, 2]\) | \(172032\) | \(1.6573\) | |
32448.be4 | 32448ch1 | \([0, -1, 0, 5183, 304225]\) | \(12167/39\) | \(-49347441721344\) | \([2]\) | \(86016\) | \(1.3107\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32448.be have rank \(0\).
Complex multiplication
The elliptic curves in class 32448.be do not have complex multiplication.Modular form 32448.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.