## The modular curve $X_{284}$

Curve name $X_{284}$
Index $48$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 9 & 2 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 9 & 9 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $24$ $X_{90}$
Meaning/Special name
Chosen covering $X_{90}$
Curves that $X_{284}$ minimally covers $X_{90}$, $X_{113}$, $X_{155}$
Curves that minimally cover $X_{284}$ $X_{580}$, $X_{584}$, $X_{650}$, $X_{674}$, $X_{695}$, $X_{706}$
Curves that minimally cover $X_{284}$ and have infinitely many rational points.
Model $y^2 = x^3 + 8x$
Info about rational points $X_{284}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 27613028724215x + 55849486008698159294$, with conductor $2094789268$
Generic density of odd order reductions $45667/172032$