## The modular curve $X_{168}$

Curve name $X_{168}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{168}$ minimally covers $X_{36}$
Curves that minimally cover $X_{168}$ $X_{305}$, $X_{313}$, $X_{330}$, $X_{332}$, $X_{335}$, $X_{336}$, $X_{337}$, $X_{338}$, $X_{339}$, $X_{340}$, $X_{427}$, $X_{428}$, $X_{429}$, $X_{430}$, $X_{431}$, $X_{432}$, $X_{433}$, $X_{438}$
Curves that minimally cover $X_{168}$ and have infinitely many rational points.
Model $y^2 = x^3 - x$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(-1 : 0 : 1)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(1 : 0 : 1)$ $\infty$
Elliptic curve whose $2$-adic image is the subgroup None