## The modular curve $X_{86}$

Curve name $X_{86}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{86}$ minimally covers $X_{36}$
Curves that minimally cover $X_{86}$ $X_{184}$, $X_{197}$, $X_{201}$, $X_{205}$, $X_{212}$, $X_{229}$, $X_{337}$, $X_{345}$, $X_{86a}$, $X_{86b}$, $X_{86c}$, $X_{86d}$, $X_{86e}$, $X_{86f}$, $X_{86g}$, $X_{86h}$, $X_{86i}$, $X_{86j}$, $X_{86k}$, $X_{86l}$, $X_{86m}$, $X_{86n}$, $X_{86o}$, $X_{86p}$
Curves that minimally cover $X_{86}$ and have infinitely many rational points. $X_{197}$, $X_{205}$, $X_{212}$, $X_{229}$, $X_{86a}$, $X_{86b}$, $X_{86c}$, $X_{86d}$, $X_{86e}$, $X_{86f}$, $X_{86g}$, $X_{86h}$, $X_{86i}$, $X_{86j}$, $X_{86k}$, $X_{86l}$, $X_{86m}$, $X_{86n}$, $X_{86o}$, $X_{86p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{86}) = \mathbb{Q}(f_{86}), f_{36} = \frac{-8}{f_{86}^{2} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 34884x + 2462449$, with conductor $1989$
Generic density of odd order reductions $83/672$