## The modular curve $X_{165}$

Curve name $X_{165}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 13 & 10 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 13 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 15 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 11 & 11 \\ 2 & 1 \end{matrix}\right]$
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 $X_{165}$ minimally covers $X_{39}$
Curves that minimally cover $X_{165}$ $X_{288}$, $X_{308}$, $X_{358}$, $X_{359}$, $X_{381}$, $X_{390}$, $X_{395}$, $X_{399}$, $X_{400}$, $X_{402}$
Curves that minimally cover $X_{165}$ and have infinitely many rational points. $X_{288}$, $X_{308}$
Model $y^2 = x^3 + x^2 - 13x - 21$
Info about rational points $X_{165}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup None. All the rational points lift to covering modular curves.
Generic density of odd order reductions N/A