The modular curve $X_{213i}$

Curve name $X_{213i}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13h}$
$8$ $48$ $X_{102p}$
Meaning/Special name
Chosen covering $X_{213}$
Curves that $X_{213i}$ minimally covers
Curves that minimally cover $X_{213i}$
Curves that minimally cover $X_{213i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{16} - 216t^{14} - 324t^{12} + 216t^{10} + 270t^{8} + 216t^{6} - 324t^{4} - 216t^{2} - 27\] \[B(t) = 54t^{24} + 648t^{22} + 2268t^{20} + 1512t^{18} - 3078t^{16} - 3888t^{14} + 1512t^{12} - 3888t^{10} - 3078t^{8} + 1512t^{6} + 2268t^{4} + 648t^{2} + 54\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 4890x + 129447$, with conductor $1230$
Generic density of odd order reductions $19/336$

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