## The modular curve $X_{155}$

Curve name $X_{155}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 12 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$ $8$ $12$ $X_{50}$
Meaning/Special name
Chosen covering $X_{50}$
Curves that $X_{155}$ minimally covers $X_{50}$
Curves that minimally cover $X_{155}$ $X_{284}$, $X_{318}$, $X_{328}$, $X_{350}$, $X_{411}$, $X_{418}$, $X_{425}$, $X_{426}$
Curves that minimally cover $X_{155}$ and have infinitely many rational points. $X_{284}$, $X_{318}$, $X_{328}$, $X_{350}$
Model $y^2 = x^3 - 2x$
Info about rational points $X_{155}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$
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