## The modular curve $X_{291}$

Curve name $X_{291}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 9 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{91}$
Meaning/Special name
Chosen covering $X_{91}$
Curves that $X_{291}$ minimally covers $X_{91}$, $X_{114}$, $X_{156}$
Curves that minimally cover $X_{291}$ $X_{585}$, $X_{589}$, $X_{591}$, $X_{615}$, $X_{628}$, $X_{675}$, $X_{677}$, $X_{713}$
Curves that minimally cover $X_{291}$ and have infinitely many rational points.
Model $y^2 = x^3 - 2x$
Info about rational points $X_{291}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 70411166718x + 7186776239309115$, with conductor $2665869738$
Generic density of odd order reductions $12833/57344$