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Properties

Label 257754.a
Number of curves $2$
Conductor $257754$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 257754.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.a1 257754a1 \([1, 1, 0, -49953382, -131431930220]\) \(42720468528431539/1603679551488\) \(517487662444922013745152\) \([2]\) \(77045760\) \(3.3189\) \(\Gamma_0(N)\)-optimal
257754.a2 257754a2 \([1, 1, 0, 20282778, -471585653100]\) \(2859720454700621/299395539781632\) \(-96611257457435838471395328\) \([2]\) \(154091520\) \(3.6655\)  

Rank

sage: E.rank()
 

The elliptic curves in class 257754.a have rank \(0\).

Complex multiplication

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

Modular form 257754.2.a.a

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