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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 94710cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
94710.cx4 | 94710cy1 | \([1, 0, 0, -12142535, 16280729097]\) | \(197993898174778925173824241/58723661173161600000\) | \(58723661173161600000\) | \([10]\) | \(6720000\) | \(2.7719\) | \(\Gamma_0(N)\)-optimal |
94710.cx3 | 94710cy2 | \([1, 0, 0, -13736615, 11732181225]\) | \(286657225480543563498731761/106415753000070937500000\) | \(106415753000070937500000\) | \([10]\) | \(13440000\) | \(3.1184\) | |
94710.cx2 | 94710cy3 | \([1, 0, 0, -231829835, -1356329791443]\) | \(1377944388663050233311549787441/2723783207744786460148260\) | \(2723783207744786460148260\) | \([2]\) | \(33600000\) | \(3.5766\) | |
94710.cx1 | 94710cy4 | \([1, 0, 0, -3707515865, -86890877578125]\) | \(5636023126211175233366348044223761/22086834363269135887350\) | \(22086834363269135887350\) | \([2]\) | \(67200000\) | \(3.9231\) |
Rank
sage: E.rank()
The elliptic curves in class 94710cy have rank \(1\).
Complex multiplication
The elliptic curves in class 94710cy do not have complex multiplication.Modular form 94710.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.