Properties

Label 176610.j
Number of curves 8
Conductor 176610
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("176610.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 176610.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
176610.j1 176610di8 [1, 1, 0, -1615392817, -24990620072729] [2] 51380224  
176610.j2 176610di6 [1, 1, 0, -100962067, -390509855879] [2, 2] 25690112  
176610.j3 176610di7 [1, 1, 0, -100331317, -395629149029] [2] 51380224  
176610.j4 176610di3 [1, 1, 0, -12673887, 17356463829] [2] 12845056  
176610.j5 176610di4 [1, 1, 0, -6349567, -6023578379] [2, 2] 12845056  
176610.j6 176610di2 [1, 1, 0, -899887, 192326629] [2, 2] 6422528  
176610.j7 176610di1 [1, 1, 0, 176593, 21596901] [2] 3211264 \(\Gamma_0(N)\)-optimal
176610.j8 176610di5 [1, 1, 0, 1068053, -19246227791] [2] 25690112  

Rank

sage: E.rank()
 

The elliptic curves in class 176610.j have rank \(2\).

Modular form 176610.2.a.j

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} - q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.