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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 10080.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10080.bc1 | 10080s3 | \([0, 0, 0, -10083, 389702]\) | \(303735479048/105\) | \(39191040\) | \([2]\) | \(12288\) | \(0.81289\) | |
| 10080.bc2 | 10080s2 | \([0, 0, 0, -1308, -9088]\) | \(82881856/36015\) | \(107540213760\) | \([2]\) | \(12288\) | \(0.81289\) | |
| 10080.bc3 | 10080s1 | \([0, 0, 0, -633, 6032]\) | \(601211584/11025\) | \(514382400\) | \([2, 2]\) | \(6144\) | \(0.46632\) | \(\Gamma_0(N)\)-optimal |
| 10080.bc4 | 10080s4 | \([0, 0, 0, -3, 17498]\) | \(-8/354375\) | \(-132269760000\) | \([2]\) | \(12288\) | \(0.81289\) |
Rank
sage: E.rank()
The elliptic curves in class 10080.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 10080.bc do not have complex multiplication.Modular form 10080.2.a.bc
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.
