Properties

Label 182070.bf
Number of curves $6$
Conductor $182070$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("182070.bf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 182070.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
182070.bf1 182070cw5 [1, -1, 0, -43696854, 111190080078] [2] 10485760  
182070.bf2 182070cw3 [1, -1, 0, -2731104, 1737789228] [2, 2] 5242880  
182070.bf3 182070cw6 [1, -1, 0, -2549034, 1979323290] [2] 10485760  
182070.bf4 182070cw4 [1, -1, 0, -962424, -343239660] [2] 5242880  
182070.bf5 182070cw2 [1, -1, 0, -182124, 23345280] [2, 2] 2621440  
182070.bf6 182070cw1 [1, -1, 0, 25956, 2245968] [2] 1310720 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 182070.bf have rank \(1\).

Modular form 182070.2.a.bf

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

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.