# Properties

 Label 476850.ea Number of curves 2 Conductor 476850 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("476850.ea1")

sage: E.isogeny_class()

## Elliptic curves in class 476850.ea

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
476850.ea1 476850ea2 [1, 0, 1, -84966151, 301270362698] [u'2'] 74317824 $$\Gamma_0(N)$$-optimal*
476850.ea2 476850ea1 [1, 0, 1, -6358151, 2717178698] [u'2'] 37158912 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 476850.ea2.

## Rank

sage: E.rank()

The elliptic curves in class 476850.ea have rank $$1$$.

## Modular form 476850.2.a.ea

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} - 2q^{7} - q^{8} + q^{9} + q^{11} + q^{12} + 4q^{13} + 2q^{14} + q^{16} - q^{18} - 6q^{19} + O(q^{20})$$

## 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.