## The modular curve $X_{82}$

Curve name $X_{82}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{35}$
Curves that $X_{82}$ minimally covers $X_{35}$, $X_{41}$, $X_{43}$
Curves that minimally cover $X_{82}$ $X_{261}$, $X_{263}$, $X_{82a}$, $X_{82b}$, $X_{82c}$, $X_{82d}$
Curves that minimally cover $X_{82}$ and have infinitely many rational points. $X_{82a}$, $X_{82b}$, $X_{82c}$, $X_{82d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{82}) = \mathbb{Q}(f_{82}), f_{35} = \frac{4f_{82}^{2} - 8}{f_{82}^{2} + 4f_{82} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 583x - 5213$, with conductor $19200$
Generic density of odd order reductions $401/1792$