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Properties

Label 257754c
Number of curves $3$
Conductor $257754$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 257754c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.c3 257754c1 \([1, 1, 0, 38981, -79925411]\) \(139233463487/58763045376\) \(-2764559239956896256\) \([]\) \(7698240\) \(2.2177\) \(\Gamma_0(N)\)-optimal
257754.c2 257754c2 \([1, 1, 0, -350899, 2160870901]\) \(-101566487155393/42823570577256\) \(-2014672605372687082536\) \([]\) \(23094720\) \(2.7670\)  
257754.c1 257754c3 \([1, 1, 0, -137794429, 622544561359]\) \(-6150311179917589675873/244053849830826\) \(-11481728376732910127706\) \([]\) \(69284160\) \(3.3163\)  

Rank

sage: E.rank()
 

The elliptic curves in class 257754c have rank \(1\).

Complex multiplication

The elliptic curves in class 257754c do not have complex multiplication.

Modular form 257754.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{10} + 3 q^{11} - q^{12} - 5 q^{13} - q^{14} + 3 q^{15} + q^{16} - q^{17} - q^{18} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.