Properties

Label 137904.cv
Number of curves $6$
Conductor $137904$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("137904.cv1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 137904.cv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
137904.cv1 137904p5 [0, 1, 0, -54550552, 155034754772] [4] 11010048  
137904.cv2 137904p3 [0, 1, 0, -3755912, 1899074100] [2, 2] 5505024  
137904.cv3 137904p2 [0, 1, 0, -1471032, -664561260] [2, 2] 2752512  
137904.cv4 137904p1 [0, 1, 0, -1457512, -677762188] [2] 1376256 \(\Gamma_0(N)\)-optimal
137904.cv5 137904p4 [0, 1, 0, 597528, -2383120908] [2] 5505024  
137904.cv6 137904p6 [0, 1, 0, 10480648, 12872614548] [2] 11010048  

Rank

sage: E.rank()
 

The elliptic curves in class 137904.cv have rank \(0\).

Modular form 137904.2.a.cv

sage: E.q_eigenform(10)
 
\( q + q^{3} + 2q^{5} + q^{9} + 4q^{11} + 2q^{15} + q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.