The modular curve $X_{118}$

Curve name $X_{118}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $12$ $X_{36}$
Meaning/Special name $X_0(16)$
Chosen covering $X_{36}$
Curves that $X_{118}$ minimally covers $X_{36}$
Curves that minimally cover $X_{118}$ $X_{211}$, $X_{212}$, $X_{213}$, $X_{225}$, $X_{227}$, $X_{235}$, $X_{240}$, $X_{243}$, $X_{305}$, $X_{313}$, $X_{329}$, $X_{331}$, $X_{353}$, $X_{354}$, $X_{355}$, $X_{356}$, $X_{405}$, $X_{406}$, $X_{118a}$, $X_{118b}$, $X_{118c}$, $X_{118d}$, $X_{118e}$, $X_{118f}$, $X_{118g}$, $X_{118h}$, $X_{118i}$, $X_{118j}$, $X_{118k}$, $X_{118l}$, $X_{118m}$, $X_{118n}$, $X_{118o}$, $X_{118p}$, $X_{118q}$, $X_{118r}$, $X_{118s}$, $X_{118t}$, $X_{118u}$, $X_{118v}$, $X_{118w}$, $X_{118x}$
Curves that minimally cover $X_{118}$ and have infinitely many rational points. $X_{211}$, $X_{212}$, $X_{213}$, $X_{225}$, $X_{227}$, $X_{235}$, $X_{240}$, $X_{243}$, $X_{118a}$, $X_{118b}$, $X_{118c}$, $X_{118d}$, $X_{118e}$, $X_{118f}$, $X_{118g}$, $X_{118h}$, $X_{118i}$, $X_{118j}$, $X_{118k}$, $X_{118l}$, $X_{118m}$, $X_{118n}$, $X_{118o}$, $X_{118p}$, $X_{118q}$, $X_{118r}$, $X_{118s}$, $X_{118t}$, $X_{118u}$, $X_{118v}$, $X_{118w}$, $X_{118x}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{118}) = \mathbb{Q}(f_{118}), f_{36} = -f_{118}^{2}\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 40148370x - 97905135425$, with conductor $6435$
Generic density of odd order reductions $19/168$

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