# Properties

 Label 66.a Number of curves 4 Conductor 66 CM no Rank 0 Graph # Related objects

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 structure Modular degree Optimality
66.a1 66a3 [1, 0, 1, -81, -284]  12
66.a2 66a4 [1, 0, 1, -41, -556]  24
66.a3 66a1 [1, 0, 1, -6, 4]  4 $$\Gamma_0(N)$$-optimal
66.a4 66a2 [1, 0, 1, 4, 20]  8

## Rank

sage: E.rank()

The elliptic curves in class 66.a have rank $$0$$.

## Modular form66.2.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. 