# Properties

 Label 8670.k Number of curves $2$ Conductor $8670$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("k1")

E.isogeny_class()

## Elliptic curves in class 8670.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8670.k1 8670l2 $$[1, 0, 1, -13349928, 18771756406]$$ $$10901014250685308569/1040774054400$$ $$25121755551489753600$$ $$$$ $$580608$$ $$2.7592$$
8670.k2 8670l1 $$[1, 0, 1, -772648, 338494838]$$ $$-2113364608155289/828431400960$$ $$-19996320102438666240$$ $$$$ $$290304$$ $$2.4126$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 8670.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 8670.k do not have complex multiplication.

## Modular form8670.2.a.k

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 