Properties

Label 257754.w
Number of curves $4$
Conductor $257754$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 257754.w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.w1 257754w4 \([1, 0, 1, -1833527, -955758526]\) \(14489843500598257/6246072\) \(293851960029432\) \([2]\) \(5529600\) \(2.1188\)  
257754.w2 257754w3 \([1, 0, 1, -245127, 24671266]\) \(34623662831857/14438442312\) \(679269238835716872\) \([2]\) \(5529600\) \(2.1188\)  
257754.w3 257754w2 \([1, 0, 1, -115167, -14784590]\) \(3590714269297/73410624\) \(3453667480839744\) \([2, 2]\) \(2764800\) \(1.7722\)  
257754.w4 257754w1 \([1, 0, 1, 353, -691150]\) \(103823/4386816\) \(-206381623504896\) \([2]\) \(1382400\) \(1.4256\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 257754.2.a.w

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.