## The modular curve $X_{153}$

Curve name $X_{153}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{45}$
Meaning/Special name
Chosen covering $X_{45}$
Curves that $X_{153}$ minimally covers $X_{45}$
Curves that minimally cover $X_{153}$ $X_{309}$, $X_{320}$, $X_{326}$, $X_{349}$, $X_{361}$, $X_{364}$, $X_{376}$, $X_{383}$, $X_{391}$, $X_{393}$
Curves that minimally cover $X_{153}$ and have infinitely many rational points. $X_{309}$, $X_{320}$, $X_{326}$, $X_{349}$
Model $y^2 = x^3 + x^2 - 3x + 1$
Info about rational points $X_{153}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
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