## The modular curve $X_{108}$

Curve name $X_{108}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 11 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 10 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 14 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{10}$ $8$ $12$ $X_{42}$
Meaning/Special name
Chosen covering $X_{42}$
Curves that $X_{108}$ minimally covers $X_{42}$
Curves that minimally cover $X_{108}$ $X_{238}$, $X_{108a}$, $X_{108b}$, $X_{108c}$, $X_{108d}$, $X_{108e}$, $X_{108f}$
Curves that minimally cover $X_{108}$ and have infinitely many rational points. $X_{238}$, $X_{108a}$, $X_{108b}$, $X_{108c}$, $X_{108d}$, $X_{108e}$, $X_{108f}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{108}) = \mathbb{Q}(f_{108}), f_{42} = \frac{8f_{108} + 8}{f_{108}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 31x - 8169$, with conductor $10880$
Generic density of odd order reductions $85091/344064$