The modular curve $X_{213g}$

Curve name $X_{213g}$
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} 3 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{102n}$
Meaning/Special name
Chosen covering $X_{213}$
Curves that $X_{213g}$ minimally covers
Curves that minimally cover $X_{213g}$
Curves that minimally cover $X_{213g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{16} - 864t^{14} - 1296t^{12} + 864t^{10} + 1080t^{8} + 864t^{6} - 1296t^{4} - 864t^{2} - 108\] \[B(t) = -432t^{24} - 5184t^{22} - 18144t^{20} - 12096t^{18} + 24624t^{16} + 31104t^{14} - 12096t^{12} + 31104t^{10} + 24624t^{8} - 12096t^{6} - 18144t^{4} - 5184t^{2} - 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 7262081x - 7534894881$, with conductor $16320$
Generic density of odd order reductions $299/2688$

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