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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 45414.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45414.bd1 | 45414u3 | \([1, -1, 1, -103601, -16992287]\) | \(-1167051/512\) | \(-53950037424735744\) | \([]\) | \(435456\) | \(1.9177\) | |
45414.bd2 | 45414u1 | \([1, -1, 1, -2681, 54783]\) | \(-132651/2\) | \(-32120459334\) | \([]\) | \(48384\) | \(0.81914\) | \(\Gamma_0(N)\)-optimal |
45414.bd3 | 45414u2 | \([1, -1, 1, 9934, 265033]\) | \(9261/8\) | \(-93663259417944\) | \([]\) | \(145152\) | \(1.3684\) |
Rank
sage: E.rank()
The elliptic curves in class 45414.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 45414.bd do not have complex multiplication.Modular form 45414.2.a.bd
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.