## The modular curve $X_{102}$

Curve name $X_{102}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
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 $X_{102}$ minimally covers $X_{36}$
Curves that minimally cover $X_{102}$ $X_{193}$, $X_{195}$, $X_{197}$, $X_{202}$, $X_{213}$, $X_{223}$, $X_{230}$, $X_{235}$, $X_{338}$, $X_{340}$, $X_{342}$, $X_{347}$, $X_{102a}$, $X_{102b}$, $X_{102c}$, $X_{102d}$, $X_{102e}$, $X_{102f}$, $X_{102g}$, $X_{102h}$, $X_{102i}$, $X_{102j}$, $X_{102k}$, $X_{102l}$, $X_{102m}$, $X_{102n}$, $X_{102o}$, $X_{102p}$
Curves that minimally cover $X_{102}$ and have infinitely many rational points. $X_{193}$, $X_{195}$, $X_{197}$, $X_{202}$, $X_{213}$, $X_{223}$, $X_{230}$, $X_{235}$, $X_{102a}$, $X_{102b}$, $X_{102c}$, $X_{102d}$, $X_{102e}$, $X_{102f}$, $X_{102g}$, $X_{102h}$, $X_{102i}$, $X_{102j}$, $X_{102k}$, $X_{102l}$, $X_{102m}$, $X_{102n}$, $X_{102o}$, $X_{102p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{102}) = \mathbb{Q}(f_{102}), f_{36} = \frac{f_{102}^{2}}{f_{102} - 1}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 975x + 11250$, with conductor $525$
Generic density of odd order reductions $19/168$