Properties

Label 166410.a
Number of curves $4$
Conductor $166410$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 166410.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
166410.a1 166410by3 \([1, -1, 0, -13953825, 19929326575]\) \(65202655558249/512820150\) \(2363215590957632628150\) \([2]\) \(11354112\) \(2.9294\)  
166410.a2 166410by2 \([1, -1, 0, -1473075, -172169375]\) \(76711450249/41602500\) \(191715705053350402500\) \([2, 2]\) \(5677056\) \(2.5828\)  
166410.a3 166410by1 \([1, -1, 0, -1140255, -467780099]\) \(35578826569/51600\) \(237786920996403600\) \([2]\) \(2838528\) \(2.2362\) \(\Gamma_0(N)\)-optimal
166410.a4 166410by4 \([1, -1, 0, 5682555, -1358572829]\) \(4403686064471/2721093750\) \(-12539544661919721093750\) \([2]\) \(11354112\) \(2.9294\)  

Rank

sage: E.rank()
 

The elliptic curves in class 166410.a have rank \(1\).

Complex multiplication

The elliptic curves in class 166410.a do not have complex multiplication.

Modular form 166410.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} - 2 q^{13} + 4 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.