The modular curve $X_{223e}$

Curve name $X_{223e}$
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} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \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 $X_{223}$
Curves that $X_{223e}$ minimally covers
Curves that minimally cover $X_{223e}$
Curves that minimally cover $X_{223e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1811939328t^{16} - 1811939328t^{14} - 339738624t^{12} + 28311552t^{10} + 4423680t^{8} + 442368t^{6} - 82944t^{4} - 6912t^{2} - 108\] \[B(t) = -29686813949952t^{24} - 44530220924928t^{22} - 19481971654656t^{20} - 1623497637888t^{18} + 413122166784t^{16} + 65229815808t^{14} - 3170893824t^{12} + 1019215872t^{10} + 100859904t^{8} - 6193152t^{6} - 1161216t^{4} - 41472t^{2} - 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 2177x - 36993$, with conductor $3264$
Generic density of odd order reductions $271/2688$

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