## The modular curve $X_{92}$

Curve name $X_{92}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{27}$
Meaning/Special name
Chosen covering $X_{27}$
Curves that $X_{92}$ minimally covers $X_{27}$, $X_{36}$, $X_{43}$
Curves that minimally cover $X_{92}$ $X_{195}$, $X_{205}$, $X_{207}$, $X_{259}$, $X_{278}$, $X_{305}$, $X_{306}$, $X_{371}$, $X_{92a}$, $X_{92b}$, $X_{92c}$, $X_{92d}$, $X_{92e}$, $X_{92f}$, $X_{92g}$, $X_{92h}$, $X_{92i}$, $X_{92j}$, $X_{92k}$
Curves that minimally cover $X_{92}$ and have infinitely many rational points. $X_{195}$, $X_{205}$, $X_{207}$, $X_{92a}$, $X_{92b}$, $X_{92c}$, $X_{92d}$, $X_{92e}$, $X_{92f}$, $X_{92g}$, $X_{92h}$, $X_{92i}$, $X_{92j}$, $X_{92k}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{92}) = \mathbb{Q}(f_{92}), f_{27} = \frac{f_{92}}{f_{92}^{2} - 1}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 33729345x - 401153583974$, with conductor $6435$
Generic density of odd order reductions $19/168$