Properties

Label 268770.bg
Number of curves $6$
Conductor $268770$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("268770.bg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 268770.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
268770.bg1 268770bg6 [1, 1, 1, -88873286, -322518653497] [2] 20971520  
268770.bg2 268770bg4 [1, 1, 1, -5554586, -5041079017] [2, 2] 10485760  
268770.bg3 268770bg5 [1, 1, 1, -5467886, -5205947737] [2] 20971520  
268770.bg4 268770bg3 [1, 1, 1, -1069306, 331380119] [2] 10485760  
268770.bg5 268770bg2 [1, 1, 1, -352586, -76290217] [2, 2] 5242880  
268770.bg6 268770bg1 [1, 1, 1, 17334, -4969641] [2] 2621440 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 268770.bg have rank \(1\).

Modular form 268770.2.a.bg

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

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.