Properties

Label 66.a
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 66.a

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

Rank

sage: E.rank()

The elliptic curves in class 66.a 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 LMFDB 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 LMFDB labels.