## The modular curve $X_{99}$

Curve name $X_{99}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{25}$
Curves that $X_{99}$ minimally covers $X_{25}$
Curves that minimally cover $X_{99}$ $X_{182}$, $X_{186}$, $X_{187}$, $X_{190}$, $X_{194}$, $X_{198}$, $X_{203}$, $X_{206}$, $X_{268}$, $X_{270}$, $X_{271}$, $X_{272}$, $X_{99a}$, $X_{99b}$, $X_{99c}$, $X_{99d}$, $X_{99e}$, $X_{99f}$, $X_{99g}$, $X_{99h}$, $X_{99i}$, $X_{99j}$, $X_{99k}$, $X_{99l}$, $X_{99m}$, $X_{99n}$, $X_{99o}$, $X_{99p}$
Curves that minimally cover $X_{99}$ and have infinitely many rational points. $X_{187}$, $X_{190}$, $X_{194}$, $X_{203}$, $X_{99a}$, $X_{99b}$, $X_{99c}$, $X_{99d}$, $X_{99e}$, $X_{99f}$, $X_{99g}$, $X_{99h}$, $X_{99i}$, $X_{99j}$, $X_{99k}$, $X_{99l}$, $X_{99m}$, $X_{99n}$, $X_{99o}$, $X_{99p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{99}) = \mathbb{Q}(f_{99}), f_{25} = -f_{99}^{2} - 1$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 4896x - 126293$, with conductor $1989$
Generic density of odd order reductions $83/672$