## The modular curve $X_{150}$

Curve name $X_{150}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 10 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 13 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 4 & 13 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 12 & 15 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{45}$
Meaning/Special name
Chosen covering $X_{45}$
Curves that $X_{150}$ minimally covers $X_{45}$
Curves that minimally cover $X_{150}$ $X_{312}$, $X_{323}$, $X_{364}$, $X_{373}$, $X_{376}$, $X_{384}$, $X_{392}$, $X_{397}$, $X_{401}$, $X_{403}$
Curves that minimally cover $X_{150}$ and have infinitely many rational points. $X_{312}$, $X_{323}$
Model $y^2 = x^3 + x^2 - 13x - 21$
Info about rational points $X_{150}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
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