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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 137904cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137904.n2 | 137904cy1 | \([0, -1, 0, -74923, -28443182]\) | \(-602275072000/4184843403\) | \(-323191036819056432\) | \([2]\) | \(1548288\) | \(2.0430\) | \(\Gamma_0(N)\)-optimal |
137904.n1 | 137904cy2 | \([0, -1, 0, -1931388, -1030191696]\) | \(644811009586000/1651460733\) | \(2040649095472895232\) | \([2]\) | \(3096576\) | \(2.3896\) |
Rank
sage: E.rank()
The elliptic curves in class 137904cy have rank \(0\).
Complex multiplication
The elliptic curves in class 137904cy do not have complex multiplication.Modular form 137904.2.a.cy
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.