## The modular curve $X_{83}$

Curve name $X_{83}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{26}$
Meaning/Special name
Chosen covering $X_{26}$
Curves that $X_{83}$ minimally covers $X_{22}$, $X_{26}$, $X_{29}$, $X_{39}$
Curves that minimally cover $X_{83}$ $X_{232}$, $X_{237}$, $X_{275}$, $X_{276}$, $X_{370}$, $X_{402}$
Curves that minimally cover $X_{83}$ and have infinitely many rational points. $X_{232}$, $X_{237}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{83}) = \mathbb{Q}(f_{83}), f_{26} = \frac{f_{83}}{f_{83}^{2} - \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 2110x + 682176$, with conductor $202496$
Generic density of odd order reductions $1427/5376$