# Properties

 Label 424830f Number of curves $2$ Conductor $424830$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 424830f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
424830.f1 424830f1 [1, 1, 0, -470124253, -4030659158147] [] 274827168 $$\Gamma_0(N)$$-optimal*
424830.f2 424830f2 [1, 1, 0, 2184001172, -14826048911792] [] 824481504 $$\Gamma_0(N)$$-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 424830f1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 424830.2.a.f

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels.