# Properties

 Label 666.f Number of curves $4$ Conductor $666$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("f1")

E.isogeny_class()

## Elliptic curves in class 666.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
666.f1 666f4 $$[1, -1, 1, -7241, 238677]$$ $$57588477431113/78653268$$ $$57338232372$$ $$$$ $$1152$$ $$0.96920$$
666.f2 666f3 $$[1, -1, 1, -5441, -151995]$$ $$24431916147913/202409388$$ $$147556443852$$ $$$$ $$1152$$ $$0.96920$$
666.f3 666f2 $$[1, -1, 1, -581, 1581]$$ $$29704593673/15968016$$ $$11640683664$$ $$[2, 2]$$ $$576$$ $$0.62263$$
666.f4 666f1 $$[1, -1, 1, 139, 141]$$ $$410172407/255744$$ $$-186437376$$ $$$$ $$288$$ $$0.27605$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 666.f have rank $$0$$.

## Complex multiplication

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

## Modular form666.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 2 q^{5} + q^{8} - 2 q^{10} + 4 q^{11} + 6 q^{13} + q^{16} - 6 q^{17} + 8 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 