Properties

Label 322161e
Number of curves $6$
Conductor $322161$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("322161.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 322161e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
322161.e5 322161e1 [1, 1, 1, -414747, -110245872] [2] 4866048 \(\Gamma_0(N)\)-optimal
322161.e4 322161e2 [1, 1, 1, -6765392, -6775882864] [2, 2] 9732096  
322161.e3 322161e3 [1, 1, 1, -6894997, -6502934734] [2, 2] 19464192  
322161.e1 322161e4 [1, 1, 1, -108246107, -433522585582] [2] 19464192  
322161.e2 322161e5 [1, 1, 1, -22466112, 33309292098] [2] 38928384  
322161.e6 322161e6 [1, 1, 1, 6602438, -28843889146] [2] 38928384  

Rank

sage: E.rank()
 

The elliptic curves in class 322161e have rank \(1\).

Modular form 322161.2.a.e

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

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 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.