## The modular curve $X_{161}$

Curve name $X_{161}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 5 & 15 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 11 & 11 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{39}$
Meaning/Special name
Chosen covering $X_{39}$
Curves that $X_{161}$ minimally covers $X_{39}$
Curves that minimally cover $X_{161}$ $X_{287}$, $X_{301}$, $X_{360}$, $X_{365}$, $X_{370}$, $X_{372}$, $X_{379}$, $X_{394}$, $X_{398}$, $X_{400}$
Curves that minimally cover $X_{161}$ and have infinitely many rational points.
Model $y^2 = x^3 - x^2 - 3x - 1$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(-1 : 0 : 1)$ $1728 \,\,(\text{CM by }-4)$
Elliptic curve whose $2$-adic image is the subgroup None