# Properties

 Label 424830.x Number of curves $4$ Conductor $424830$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 424830.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.x1 424830x4 $$[1, 1, 0, -5289428, -4684521582]$$ $$5763259856089/5670$$ $$16101444049443270$$ $$$$ $$14155776$$ $$2.4031$$
424830.x2 424830x2 $$[1, 1, 0, -333078, -72142272]$$ $$1439069689/44100$$ $$125233453717892100$$ $$[2, 2]$$ $$7077888$$ $$2.0566$$
424830.x3 424830x1 $$[1, 1, 0, -49858, 2684452]$$ $$4826809/1680$$ $$4770798236872080$$ $$$$ $$3538944$$ $$1.7100$$ $$\Gamma_0(N)$$-optimal*
424830.x4 424830x3 $$[1, 1, 0, 91752, -243008898]$$ $$30080231/9003750$$ $$-25568496800736303750$$ $$$$ $$14155776$$ $$2.4031$$
*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.x1.

## Rank

sage: E.rank()

The elliptic curves in class 424830.x have rank $$0$$.

## Complex multiplication

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

## Modular form 424830.2.a.x

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 