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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 68400.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.bf1 | 68400eo2 | \([0, 0, 0, -23175, -1356750]\) | \(472058064/475\) | \(1385100000000\) | \([2]\) | \(147456\) | \(1.2489\) | |
68400.bf2 | 68400eo1 | \([0, 0, 0, -1800, -10125]\) | \(3538944/1805\) | \(328961250000\) | \([2]\) | \(73728\) | \(0.90229\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68400.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 68400.bf do not have complex multiplication.Modular form 68400.2.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 LMFDB 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 LMFDB labels.