Properties

Label 166635.bi
Number of curves $4$
Conductor $166635$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("bi1")
 
E.isogeny_class()
 

Elliptic curves in class 166635.bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
166635.bi1 166635bw3 \([1, -1, 0, -219010860, 1247572088091]\) \(10765299591712341649/20708625\) \(2234836769933273625\) \([2]\) \(17842176\) \(3.2025\)  
166635.bi2 166635bw2 \([1, -1, 0, -13692735, 19482255216]\) \(2630872462131649/3645140625\) \(393376880421928265625\) \([2, 2]\) \(8921088\) \(2.8560\)  
166635.bi3 166635bw4 \([1, -1, 0, -9860130, 30619038825]\) \(-982374577874929/3183837890625\) \(-343593936703935791015625\) \([2]\) \(17842176\) \(3.2025\)  
166635.bi4 166635bw1 \([1, -1, 0, -1099890, 116978175]\) \(1363569097969/734582625\) \(79274807521219067625\) \([2]\) \(4460544\) \(2.5094\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 166635.bi have rank \(1\).

Complex multiplication

The elliptic curves in class 166635.bi do not have complex multiplication.

Modular form 166635.2.a.bi

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