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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 13680bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13680.ba4 | 13680bf1 | \([0, 0, 0, 1797, -25558]\) | \(214921799/218880\) | \(-653572177920\) | \([2]\) | \(24576\) | \(0.95399\) | \(\Gamma_0(N)\)-optimal |
13680.ba3 | 13680bf2 | \([0, 0, 0, -9723, -235222]\) | \(34043726521/11696400\) | \(34925263257600\) | \([2, 2]\) | \(49152\) | \(1.3006\) | |
13680.ba1 | 13680bf3 | \([0, 0, 0, -139323, -20012182]\) | \(100162392144121/23457780\) | \(70044555755520\) | \([2]\) | \(98304\) | \(1.6471\) | |
13680.ba2 | 13680bf4 | \([0, 0, 0, -64443, 6123242]\) | \(9912050027641/311647500\) | \(930574448640000\) | \([2]\) | \(98304\) | \(1.6471\) |
Rank
sage: E.rank()
The elliptic curves in class 13680bf have rank \(0\).
Complex multiplication
The elliptic curves in class 13680bf do not have complex multiplication.Modular form 13680.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.