Properties

Label 6699b
Number of curves $4$
Conductor $6699$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6699b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6699.d3 6699b1 \([1, 1, 0, -49, 112]\) \(13430356633/180873\) \(180873\) \([2]\) \(832\) \(-0.18454\) \(\Gamma_0(N)\)-optimal
6699.d2 6699b2 \([1, 1, 0, -94, -185]\) \(93391282153/44876601\) \(44876601\) \([2, 2]\) \(1664\) \(0.16203\)  
6699.d1 6699b3 \([1, 1, 0, -1249, -17510]\) \(215751695207833/163381911\) \(163381911\) \([2]\) \(3328\) \(0.50861\)  
6699.d4 6699b4 \([1, 1, 0, 341, -968]\) \(4365111505607/3058314567\) \(-3058314567\) \([2]\) \(3328\) \(0.50861\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6699b have rank \(0\).

Complex multiplication

The elliptic curves in class 6699b do not have complex multiplication.

Modular form 6699.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - q^{7} - 3q^{8} + q^{9} + 2q^{10} + q^{11} + q^{12} - 2q^{13} - q^{14} - 2q^{15} - q^{16} - 2q^{17} + q^{18} + O(q^{20})\)  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 Cremona 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 Cremona labels.