Properties

Label 485520.b
Number of curves $6$
Conductor $485520$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("485520.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 485520.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
485520.b1 485520b5 [0, -1, 0, -3028816, 2029849696] [2] 10485760 \(\Gamma_0(N)\)-optimal*
485520.b2 485520b3 [0, -1, 0, -196616, 29183616] [2, 2] 5242880 \(\Gamma_0(N)\)-optimal*
485520.b3 485520b2 [0, -1, 0, -52116, -4109184] [2, 2] 2621440 \(\Gamma_0(N)\)-optimal*
485520.b4 485520b1 [0, -1, 0, -50671, -4373330] [2] 1310720 \(\Gamma_0(N)\)-optimal*
485520.b5 485520b4 [0, -1, 0, 69264, -20519760] [2] 5242880  
485520.b6 485520b6 [0, -1, 0, 323584, 156944736] [2] 10485760  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 485520.b4.

Rank

sage: E.rank()
 

The elliptic curves in class 485520.b have rank \(1\).

Modular form 485520.2.a.b

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

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.