Properties

Label 116610.bb
Number of curves $6$
Conductor $116610$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 116610.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116610.bb1 116610y4 \([1, 0, 1, -18657604, 31017738806]\) \(148809678420065817601/20700\) \(99914946300\) \([2]\) \(3538944\) \(2.4355\)  
116610.bb2 116610y6 \([1, 0, 1, -4365274, -3007392178]\) \(1905890658841300321/293666194803750\) \(1417470632074493733750\) \([2]\) \(7077888\) \(2.7821\)  
116610.bb3 116610y3 \([1, 0, 1, -1196524, 457952822]\) \(39248884582600321/3935264062500\) \(18994767994251562500\) \([2, 2]\) \(3538944\) \(2.4355\)  
116610.bb4 116610y2 \([1, 0, 1, -1166104, 484576406]\) \(36330796409313601/428490000\) \(2068239388410000\) \([2, 2]\) \(1769472\) \(2.0889\)  
116610.bb5 116610y1 \([1, 0, 1, -70984, 7980182]\) \(-8194759433281/965779200\) \(-4661631734572800\) \([2]\) \(884736\) \(1.7423\) \(\Gamma_0(N)\)-optimal
116610.bb6 116610y5 \([1, 0, 1, 1485506, 2219510126]\) \(75108181893694559/484313964843750\) \(-2337691004333496093750\) \([2]\) \(7077888\) \(2.7821\)  

Rank

sage: E.rank()
 

The elliptic curves in class 116610.bb have rank \(2\).

Complex multiplication

The elliptic curves in class 116610.bb do not have complex multiplication.

Modular form 116610.2.a.bb

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.