## The modular curve $X_{149}$

Curve name $X_{149}$
Index $24$
Level $16$
Genus $1$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 10 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 13 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 4 & 13 \end{matrix}\right]$
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 $X_{149}$ minimally covers $X_{45}$
Curves that minimally cover $X_{149}$ $X_{311}$, $X_{322}$, $X_{325}$, $X_{351}$, $X_{363}$, $X_{367}$, $X_{378}$, $X_{385}$, $X_{391}$, $X_{392}$
Curves that minimally cover $X_{149}$ and have infinitely many rational points.
Model $y^2 = x^3 - x^2 - 13x + 21$
 Rational point Image on the $j$-line $(0 : 1 : 0)$ $\infty$ $(3 : 0 : 1)$ $1728 \,\,(\text{CM by }-4)$
Elliptic curve whose $2$-adic image is the subgroup None