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Properties

Label 128226.s
Number of curves $3$
Conductor $128226$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 128226.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
128226.s1 128226ba3 \([1, 0, 0, -7593771927, -254703853363449]\) \(48427980631254958469835824847167473/153101623358112538524114\) \(153101623358112538524114\) \([]\) \(151165440\) \(4.0983\)  
128226.s2 128226ba2 \([1, 0, 0, -97056897, -323425585791]\) \(101112050932948721991382242193/13327203141829750320519624\) \(13327203141829750320519624\) \([3]\) \(50388480\) \(3.5490\)  
128226.s3 128226ba1 \([1, 0, 0, -24080217, 45421448121]\) \(1544204814149745316374461713/2295658741816117891584\) \(2295658741816117891584\) \([9]\) \(16796160\) \(2.9997\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 128226.s have rank \(1\).

Complex multiplication

The elliptic curves in class 128226.s do not have complex multiplication.

Modular form 128226.2.a.s

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 3 q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 3 q^{10} - 6 q^{11} + q^{12} - 4 q^{13} + q^{14} - 3 q^{15} + q^{16} + q^{18} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.