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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 45864bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45864.be1 | 45864bm1 | \([0, 0, 0, -1470, -9947]\) | \(256000/117\) | \(160554178512\) | \([2]\) | \(36864\) | \(0.84509\) | \(\Gamma_0(N)\)-optimal |
45864.be2 | 45864bm2 | \([0, 0, 0, 5145, -74774]\) | \(686000/507\) | \(-11131756376832\) | \([2]\) | \(73728\) | \(1.1917\) |
Rank
sage: E.rank()
The elliptic curves in class 45864bm have rank \(1\).
Complex multiplication
The elliptic curves in class 45864bm do not have complex multiplication.Modular form 45864.2.a.bm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.