## The modular curve $X_{75}$

Curve name $X_{75}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{34}$
Curves that $X_{75}$ minimally covers $X_{34}$, $X_{36}$, $X_{48}$
Curves that minimally cover $X_{75}$ $X_{197}$, $X_{199}$, $X_{217}$, $X_{329}$, $X_{330}$, $X_{75a}$, $X_{75b}$, $X_{75c}$, $X_{75d}$, $X_{75e}$, $X_{75f}$, $X_{75g}$, $X_{75h}$, $X_{75i}$, $X_{75j}$
Curves that minimally cover $X_{75}$ and have infinitely many rational points. $X_{197}$, $X_{199}$, $X_{217}$, $X_{75a}$, $X_{75b}$, $X_{75c}$, $X_{75d}$, $X_{75e}$, $X_{75f}$, $X_{75g}$, $X_{75h}$, $X_{75i}$, $X_{75j}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{75}) = \mathbb{Q}(f_{75}), f_{34} = \frac{f_{75}}{f_{75}^{2} + \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 125421x + 5866702$, with conductor $5544$
Generic density of odd order reductions $643/5376$