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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 333270cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.cy1 | 333270cy1 | \([1, -1, 1, -2606172293, 21411682137981]\) | \(489781415227546051766883/233890092903563264000\) | \(934851451444332634457505792000\) | \([2]\) | \(544997376\) | \(4.4447\) | \(\Gamma_0(N)\)-optimal |
333270.cy2 | 333270cy2 | \([1, -1, 1, 9354475387, 162848733083517]\) | \(22649115256119592694355357/15973509811739648000000\) | \(-63845623586639738562130944000000\) | \([2]\) | \(1089994752\) | \(4.7913\) |
Rank
sage: E.rank()
The elliptic curves in class 333270cy have rank \(1\).
Complex multiplication
The elliptic curves in class 333270cy do not have complex multiplication.Modular form 333270.2.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.