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
LevelIndex of imageCorresponding 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 \[y^2 = x^3 - x^2 - 13x + 21\]
Info about rational points
Rational pointImage on the $j$-line
$(0 : 1 : 0)$ \[ \infty \]
$(3 : 0 : 1)$ \[1728 \,\,(\text{CM by }-4)\]
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup None
Generic density of odd order reductions N/A

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