# Properties

 Label 666.a Number of curves $2$ Conductor $666$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("a1")

E.isogeny_class()

## Elliptic curves in class 666.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
666.a1 666c2 $$[1, -1, 0, -1332, 19062]$$ $$-358667682625/303918$$ $$-221556222$$ $$$$ $$288$$ $$0.52868$$
666.a2 666c1 $$[1, -1, 0, 18, 108]$$ $$857375/7992$$ $$-5826168$$ $$[]$$ $$96$$ $$-0.020627$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 666.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 666.a do not have complex multiplication.

## Modular form666.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{7} - q^{8} - 3 q^{11} - q^{13} + q^{14} + q^{16} + 3 q^{17} - 7 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 