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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 33810bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.bb2 | 33810bf1 | \([1, 0, 1, -24869, 1502876]\) | \(14457238157881/49990500\) | \(5881332334500\) | \([2]\) | \(110592\) | \(1.3132\) | \(\Gamma_0(N)\)-optimal |
33810.bb1 | 33810bf2 | \([1, 0, 1, -36139, 1712]\) | \(44365623586201/25674468750\) | \(3020575573968750\) | \([2]\) | \(221184\) | \(1.6598\) |
Rank
sage: E.rank()
The elliptic curves in class 33810bf have rank \(1\).
Complex multiplication
The elliptic curves in class 33810bf do not have complex multiplication.Modular form 33810.2.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.