## The modular curve $X_{76}$

Curve name $X_{76}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{23}$
Meaning/Special name
Chosen covering $X_{23}$
Curves that $X_{76}$ minimally covers $X_{23}$, $X_{29}$, $X_{49}$
Curves that minimally cover $X_{76}$ $X_{257}$, $X_{258}$, $X_{76a}$, $X_{76b}$, $X_{76c}$, $X_{76d}$
Curves that minimally cover $X_{76}$ and have infinitely many rational points. $X_{76a}$, $X_{76b}$, $X_{76c}$, $X_{76d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{76}) = \mathbb{Q}(f_{76}), f_{23} = \frac{2f_{76}^{2} + 8}{f_{76}^{2} + 4f_{76} - 4}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 333x - 9963$, with conductor $19200$
Generic density of odd order reductions $401/1792$