# Properties

 Label 66.b Number of curves $4$ Conductor $66$ CM no Rank $0$ Graph # Learn more

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 $$[1, 1, 1, -352, -2689]$$ $$4824238966273/66$$ $$66$$ $$$$ $$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$$ $$$$ $$16$$ $$-0.094954$$
66.b4 66b1 $$[1, 1, 1, -2, -1]$$ $$912673/528$$ $$528$$ $$$$ $$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 form66.2.a.b

sage: E.q_eigenform(10)

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