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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 338100bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
338100.bf1 | 338100bf1 | \([0, -1, 0, -26133, 1598262]\) | \(67108864/1725\) | \(50736131250000\) | \([2]\) | \(1105920\) | \(1.4120\) | \(\Gamma_0(N)\)-optimal |
338100.bf2 | 338100bf2 | \([0, -1, 0, 4492, 5089512]\) | \(21296/23805\) | \(-11202537780000000\) | \([2]\) | \(2211840\) | \(1.7586\) |
Rank
sage: E.rank()
The elliptic curves in class 338100bf have rank \(2\).
Complex multiplication
The elliptic curves in class 338100bf do not have complex multiplication.Modular form 338100.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.