Properties

Label 45570.bo
Number of curves $6$
Conductor $45570$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("45570.bo1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 45570.bo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
45570.bo1 45570bu6 [1, 1, 1, -15068481, 22507704099] [2] 1572864  
45570.bo2 45570bu4 [1, 1, 1, -941781, 351387819] [2, 2] 786432  
45570.bo3 45570bu5 [1, 1, 1, -927081, 362906739] [2] 1572864  
45570.bo4 45570bu3 [1, 1, 1, -181301, -23241877] [2] 786432  
45570.bo5 45570bu2 [1, 1, 1, -59781, 5291019] [2, 2] 393216  
45570.bo6 45570bu1 [1, 1, 1, 2939, 348683] [2] 196608 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 45570.bo have rank \(2\).

Modular form 45570.2.a.bo

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 6q^{13} + q^{15} + q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.