# Properties

 Label 476850ds Number of curves $2$ Conductor $476850$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ds1")

sage: E.isogeny_class()

## Elliptic curves in class 476850ds

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
476850.ds2 476850ds1 [1, 0, 1, -86851, 7977098] [2] 3538944 $$\Gamma_0(N)$$-optimal*
476850.ds1 476850ds2 [1, 0, 1, -1315101, 580341598] [2] 7077888 $$\Gamma_0(N)$$-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 476850ds1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 476850.2.a.ds

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.