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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 32448.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32448.bf1 | 32448d2 | \([0, -1, 0, -1961977, -1056798887]\) | \(42246001231552/14414517\) | \(284983789102092288\) | \([2]\) | \(516096\) | \(2.3208\) | |
32448.bf2 | 32448d1 | \([0, -1, 0, -105512, -21262710]\) | \(-420526439488/390971529\) | \(-120777273274941504\) | \([2]\) | \(258048\) | \(1.9743\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32448.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 32448.bf do not have complex multiplication.Modular form 32448.2.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.