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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2299.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2299.b1 | 2299d3 | \([0, 1, 1, -93089, 10900929]\) | \(-50357871050752/19\) | \(-33659659\) | \([]\) | \(4050\) | \(1.2324\) | |
2299.b2 | 2299d2 | \([0, 1, 1, -1129, 15164]\) | \(-89915392/6859\) | \(-12151136899\) | \([]\) | \(1350\) | \(0.68308\) | |
2299.b3 | 2299d1 | \([0, 1, 1, 81, 39]\) | \(32768/19\) | \(-33659659\) | \([]\) | \(450\) | \(0.13377\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2299.b have rank \(0\).
Complex multiplication
The elliptic curves in class 2299.b do not have complex multiplication.Modular form 2299.2.a.b
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.