# Properties

 Label 1710.s Number of curves $2$ Conductor $1710$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 1710.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.s1 1710r2 $$[1, -1, 1, -14567, -673041]$$ $$468898230633769/5540400$$ $$4038951600$$ $$$$ $$3072$$ $$0.99330$$
1710.s2 1710r1 $$[1, -1, 1, -887, -10929]$$ $$-105756712489/12476160$$ $$-9095120640$$ $$$$ $$1536$$ $$0.64672$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1710.s have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1710.s do not have complex multiplication.

## Modular form1710.2.a.s

sage: E.q_eigenform(10)

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