Properties

Label 66.b
Number of curves $4$
Conductor $66$
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 66.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
66.b1 66b3 \([1, 1, 1, -352, -2689]\) \(4824238966273/66\) \(66\) \([2]\) \(16\) \(-0.094954\)  
66.b2 66b2 \([1, 1, 1, -22, -49]\) \(1180932193/4356\) \(4356\) \([2, 2]\) \(8\) \(-0.44153\)  
66.b3 66b4 \([1, 1, 1, -12, -81]\) \(-192100033/2371842\) \(-2371842\) \([2]\) \(16\) \(-0.094954\)  
66.b4 66b1 \([1, 1, 1, -2, -1]\) \(912673/528\) \(528\) \([4]\) \(4\) \(-0.78810\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 66.b have rank \(0\).

Complex multiplication

The elliptic curves in class 66.b do not have complex multiplication.

Modular form 66.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} - 4q^{7} + q^{8} + q^{9} + 2q^{10} - q^{11} - q^{12} - 6q^{13} - 4q^{14} - 2q^{15} + q^{16} + 2q^{17} + q^{18} + 4q^{19} + O(q^{20})\)  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 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.