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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 20286bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20286.d2 | 20286bf1 | \([1, -1, 0, 1314, 80716]\) | \(2924207/34776\) | \(-2982602623896\) | \([]\) | \(46080\) | \(1.0746\) | \(\Gamma_0(N)\)-optimal |
20286.d1 | 20286bf2 | \([1, -1, 0, -11916, -2282162]\) | \(-2181825073/25039686\) | \(-2147556739278006\) | \([]\) | \(138240\) | \(1.6239\) |
Rank
sage: E.rank()
The elliptic curves in class 20286bf have rank \(2\).
Complex multiplication
The elliptic curves in class 20286bf do not have complex multiplication.Modular form 20286.2.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.