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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 159936.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.bc1 | 159936cz4 | \([0, -1, 0, -106689, -12154911]\) | \(17418812548/1753941\) | \(13523314587009024\) | \([2]\) | \(983040\) | \(1.8314\) | |
159936.bc2 | 159936cz2 | \([0, -1, 0, -24369, 1263249]\) | \(830321872/127449\) | \(245665749417984\) | \([2, 2]\) | \(491520\) | \(1.4848\) | |
159936.bc3 | 159936cz1 | \([0, -1, 0, -23389, 1384573]\) | \(11745974272/357\) | \(43008709632\) | \([2]\) | \(245760\) | \(1.1383\) | \(\Gamma_0(N)\)-optimal |
159936.bc4 | 159936cz3 | \([0, -1, 0, 42271, 6900993]\) | \(1083360092/3306177\) | \(-25491434233724928\) | \([2]\) | \(983040\) | \(1.8314\) |
Rank
sage: E.rank()
The elliptic curves in class 159936.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 159936.bc do not have complex multiplication.Modular form 159936.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.