Properties

Label 8400.bn
Number of curves $6$
Conductor $8400$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("8400.bn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8400.bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8400.bn1 8400cd5 [0, 1, 0, -313608, 67492788] [2] 32768  
8400.bn2 8400cd4 [0, 1, 0, -19608, 1048788] [2, 2] 16384  
8400.bn3 8400cd3 [0, 1, 0, -15608, -751212] [2] 16384  
8400.bn4 8400cd6 [0, 1, 0, -13608, 1708788] [2] 32768  
8400.bn5 8400cd2 [0, 1, 0, -1608, 4788] [2, 2] 8192  
8400.bn6 8400cd1 [0, 1, 0, 392, 788] [2] 4096 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8400.bn have rank \(1\).

Modular form 8400.2.a.bn

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{7} + q^{9} - 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.