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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 18240bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18240.cy2 | 18240bp1 | \([0, 1, 0, -6305, 209343]\) | \(-105756712489/12476160\) | \(-3270550487040\) | \([2]\) | \(36864\) | \(1.1371\) | \(\Gamma_0(N)\)-optimal |
18240.cy1 | 18240bp2 | \([0, 1, 0, -103585, 12797375]\) | \(468898230633769/5540400\) | \(1452382617600\) | \([2]\) | \(73728\) | \(1.4837\) |
Rank
sage: E.rank()
The elliptic curves in class 18240bp have rank \(0\).
Complex multiplication
The elliptic curves in class 18240bp do not have complex multiplication.Modular form 18240.2.a.bp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.