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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 257754.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
257754.c1 | 257754c3 | \([1, 1, 0, -137794429, 622544561359]\) | \(-6150311179917589675873/244053849830826\) | \(-11481728376732910127706\) | \([]\) | \(69284160\) | \(3.3163\) | |
257754.c2 | 257754c2 | \([1, 1, 0, -350899, 2160870901]\) | \(-101566487155393/42823570577256\) | \(-2014672605372687082536\) | \([]\) | \(23094720\) | \(2.7670\) | |
257754.c3 | 257754c1 | \([1, 1, 0, 38981, -79925411]\) | \(139233463487/58763045376\) | \(-2764559239956896256\) | \([]\) | \(7698240\) | \(2.2177\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 257754.c have rank \(1\).
Complex multiplication
The elliptic curves in class 257754.c do not have complex multiplication.Modular form 257754.2.a.c
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.