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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 45486.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45486.s1 | 45486t4 | \([1, -1, 0, -58974291, 174332632099]\) | \(661397832743623417/443352042\) | \(15205399921189432458\) | \([2]\) | \(3686400\) | \(2.9968\) | |
45486.s2 | 45486t2 | \([1, -1, 0, -3708801, 2689073257]\) | \(164503536215257/4178071044\) | \(143292993163120357956\) | \([2, 2]\) | \(1843200\) | \(2.6503\) | |
45486.s3 | 45486t1 | \([1, -1, 0, -524781, -85481771]\) | \(466025146777/177366672\) | \(6083046709978685328\) | \([2]\) | \(921600\) | \(2.3037\) | \(\Gamma_0(N)\)-optimal |
45486.s4 | 45486t3 | \([1, -1, 0, 612369, 8578827967]\) | \(740480746823/927484650666\) | \(-31809428395823661717834\) | \([2]\) | \(3686400\) | \(2.9968\) |
Rank
sage: E.rank()
The elliptic curves in class 45486.s have rank \(0\).
Complex multiplication
The elliptic curves in class 45486.s do not have complex multiplication.Modular form 45486.2.a.s
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.