Properties

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

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 166410.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
166410.q1 166410cj4 \([1, -1, 0, -402504, 25793810]\) \(57960603/31250\) \(3888230902920843750\) \([2]\) \(3773952\) \(2.2581\)  
166410.q2 166410cj2 \([1, -1, 0, -236094, -44094692]\) \(8527173507/200\) \(34135360464600\) \([2]\) \(1257984\) \(1.7088\)  
166410.q3 166410cj1 \([1, -1, 0, -14214, -739340]\) \(-1860867/320\) \(-54616576743360\) \([2]\) \(628992\) \(1.3622\) \(\Gamma_0(N)\)-optimal
166410.q4 166410cj3 \([1, -1, 0, 96726, 3128768]\) \(804357/500\) \(-62211694446733500\) \([2]\) \(1886976\) \(1.9115\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 166410.2.a.q

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.