# Properties

 Label 424830.s Number of curves $2$ Conductor $424830$ CM no Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 424830.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.s1 424830s2 $$[1, 1, 0, -10875981078, -436571405239668]$$ $$-12242088317612041/600$$ $$-6973082191981377389400$$ $$[]$$ $$349780032$$ $$4.1151$$
424830.s2 424830s1 $$[1, 1, 0, -133092453, -609944793093]$$ $$-22434194041/843750$$ $$-9805896832473811953843750$$ $$[]$$ $$116593344$$ $$3.5658$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 424830.s1.

## Rank

sage: E.rank()

The elliptic curves in class 424830.s have rank $$1$$.

## Complex multiplication

The elliptic curves in class 424830.s do not have complex multiplication.

## Modular form 424830.2.a.s

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 