## The modular curve $X_{159}$

Curve name $X_{159}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{159}$ minimally covers $X_{36}$
Curves that minimally cover $X_{159}$ $X_{306}$, $X_{315}$, $X_{329}$, $X_{331}$, $X_{341}$, $X_{342}$, $X_{344}$, $X_{345}$, $X_{346}$, $X_{347}$
Curves that minimally cover $X_{159}$ and have infinitely many rational points.
Model $y^2 = x^3 - 4x$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(-2 : 0 : 1)$ $\infty$ $(0 : 0 : 1)$ $\infty$ $(2 : 0 : 1)$ $\infty$
Elliptic curve whose $2$-adic image is the subgroup None