Properties

Label 266805bn
Number of curves $6$
Conductor $266805$
CM no
Rank $1$
Graph

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

Elliptic curves in class 266805bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
266805.bn6 266805bn1 [1, -1, 1, 1866523, 285120242124] [2] 44236800 \(\Gamma_0(N)\)-optimal
266805.bn5 266805bn2 [1, -1, 1, -638732282, 6103294828656] [2, 2] 88473600  
266805.bn2 266805bn3 [1, -1, 1, -10169273687, 394716838942374] [2, 2] 176947200  
266805.bn4 266805bn4 [1, -1, 1, -1357771757, -10154762932674] [2] 176947200  
266805.bn1 266805bn5 [1, -1, 1, -162708362312, 25261761097860774] [2] 353894400  
266805.bn3 266805bn6 [1, -1, 1, -10118847542, 398825036805066] [2] 353894400  

Rank

sage: E.rank()
 

The elliptic curves in class 266805bn have rank \(1\).

Complex multiplication

The elliptic curves in class 266805bn do not have complex multiplication.

Modular form 266805.2.a.bn

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.