# Properties

 Label 424830.e Number of curves $2$ Conductor $424830$ CM no Rank $0$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 424830.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.e1 424830e2 $$[1, 1, 0, -10000043, -12173472303]$$ $$557827933667759/126562500$$ $$25091935885110937500$$ $$$$ $$29360128$$ $$2.7145$$ $$\Gamma_0(N)$$-optimal*
424830.e2 424830e1 $$[1, 1, 0, -553823, -235339467]$$ $$-94756448879/65610000$$ $$-13007659562841510000$$ $$$$ $$14680064$$ $$2.3679$$ $$\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 2 curves highlighted, and conditionally curve 424830.e1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 424830.2.a.e

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 