The modular curve $X_{223f}$

Curve name $X_{223f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13f}$
$8$ $48$ $X_{102l}$
Meaning/Special name
Chosen covering $X_{223}$
Curves that $X_{223f}$ minimally covers
Curves that minimally cover $X_{223f}$
Curves that minimally cover $X_{223f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1855425871872t^{24} - 4638564679680t^{22} - 4232690270208t^{20} - 1637993152512t^{18} - 202484219904t^{16} + 15854469120t^{14} + 3878682624t^{12} + 247726080t^{10} - 49434624t^{8} - 6248448t^{6} - 252288t^{4} - 4320t^{2} - 27\] \[B(t) = -972777519512027136t^{36} - 3647915698170101760t^{34} - 5608670385936531456t^{32} - 4499096027743125504t^{30} - 1957904753627234304t^{28} - 413952933718130688t^{26} - 16105096567848960t^{24} + 8371681533886464t^{22} + 1285462236856320t^{20} + 79174500876288t^{18} + 20085347450880t^{16} + 2043867561984t^{14} - 61436067840t^{12} - 24673517568t^{10} - 1823440896t^{8} - 65470464t^{6} - 1275264t^{4} - 12960t^{2} - 54\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 157312x - 22325068$, with conductor $13872$
Generic density of odd order reductions $299/2688$

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