Properties

Label 66a
Number of curves 4
Conductor 66
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("66.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 66a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
66.a3 66a1 [1, 0, 1, -6, 4] [6] 4 \(\Gamma_0(N)\)-optimal
66.a4 66a2 [1, 0, 1, 4, 20] [6] 8  
66.a1 66a3 [1, 0, 1, -81, -284] [2] 12  
66.a2 66a4 [1, 0, 1, -41, -556] [2] 24  

Rank

sage: E.rank()
 

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

Modular form 66.2.a.a

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.