# Properties

 Label 476850gj Number of curves $2$ Conductor $476850$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gj1")

sage: E.isogeny_class()

## Elliptic curves in class 476850gj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
476850.gj2 476850gj1 [1, 1, 1, 191312, 28264781] [2] 7077888 $$\Gamma_0(N)$$-optimal*
476850.gj1 476850gj2 [1, 1, 1, -1036938, 259175781] [2] 14155776 $$\Gamma_0(N)$$-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 476850gj1.

## Rank

sage: E.rank()

The elliptic curves in class 476850gj have rank $$0$$.

## Complex multiplication

The elliptic curves in class 476850gj do not have complex multiplication.

## Modular form 476850.2.a.gj

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels.