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Properties

Label 257754.m
Number of curves $2$
Conductor $257754$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 257754.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.m1 257754m2 \([1, 1, 0, -620451790, 5948278612342]\) \(-561469581977282220768625/208053100458\) \(-9788041405828113498\) \([]\) \(50388480\) \(3.4332\)  
257754.m2 257754m1 \([1, 1, 0, -7527940, 8451412504]\) \(-1002837679918908625/76015542235752\) \(-3576218154173662537512\) \([]\) \(16796160\) \(2.8839\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 257754.2.a.m

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.