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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 20808bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20808.p2 | 20808bf1 | \([0, 0, 0, -255, 578]\) | \(2000\) | \(916883712\) | \([2]\) | \(6144\) | \(0.41253\) | \(\Gamma_0(N)\)-optimal |
20808.p1 | 20808bf2 | \([0, 0, 0, -3315, 73406]\) | \(1098500\) | \(3667534848\) | \([2]\) | \(12288\) | \(0.75910\) |
Rank
sage: E.rank()
The elliptic curves in class 20808bf have rank \(1\).
Complex multiplication
The elliptic curves in class 20808bf do not have complex multiplication.Modular form 20808.2.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.