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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 25872bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25872.d1 | 25872bf1 | \([0, -1, 0, -9032, -485904]\) | \(-3451273/2376\) | \(-56103596752896\) | \([]\) | \(72576\) | \(1.3385\) | \(\Gamma_0(N)\)-optimal |
25872.d2 | 25872bf2 | \([0, -1, 0, 73288, 7285104]\) | \(1843623047/2044416\) | \(-48274028139380736\) | \([]\) | \(217728\) | \(1.8878\) |
Rank
sage: E.rank()
The elliptic curves in class 25872bf have rank \(1\).
Complex multiplication
The elliptic curves in class 25872bf do not have complex multiplication.Modular form 25872.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.