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Properties

Label 166782.l
Number of curves $4$
Conductor $166782$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 166782.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
166782.l1 166782cl4 \([1, 1, 0, -5102020, 4433255074]\) \(312196988566716625/25367712678\) \(1193446391891379318\) \([2]\) \(4105728\) \(2.5138\)  
166782.l2 166782cl3 \([1, 1, 0, -297110, 79045632]\) \(-61653281712625/21875235228\) \(-1029139713383495868\) \([2]\) \(2052864\) \(2.1672\)  
166782.l3 166782cl2 \([1, 1, 0, -131050, -9213092]\) \(5290763640625/2291573592\) \(107809098511974552\) \([2]\) \(1368576\) \(1.9645\)  
166782.l4 166782cl1 \([1, 1, 0, 27790, -1048716]\) \(50447927375/39517632\) \(-1859141812473792\) \([2]\) \(684288\) \(1.6179\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 166782.l have rank \(1\).

Complex multiplication

The elliptic curves in class 166782.l do not have complex multiplication.

Modular form 166782.2.a.l

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.