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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 61710cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61710.bw2 | 61710cb1 | \([1, 1, 1, -1406325, 638125467]\) | \(173629978755828841/1000026931200\) | \(1771608710263603200\) | \([2]\) | \(1689600\) | \(2.3427\) | \(\Gamma_0(N)\)-optimal |
61710.bw1 | 61710cb2 | \([1, 1, 1, -22470005, 40987710875]\) | \(708234550511150304361/23696640000\) | \(41980043255040000\) | \([2]\) | \(3379200\) | \(2.6893\) |
Rank
sage: E.rank()
The elliptic curves in class 61710cb have rank \(1\).
Complex multiplication
The elliptic curves in class 61710cb do not have complex multiplication.Modular form 61710.2.a.cb
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.