The modular curve $X_{233f}$

Curve name $X_{233f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{78e}$
Meaning/Special name
Chosen covering $X_{233}$
Curves that $X_{233f}$ minimally covers
Curves that minimally cover $X_{233f}$
Curves that minimally cover $X_{233f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1855425871872t^{24} - 60301340835840t^{22} - 192094559797248t^{20} - 249511293222912t^{18} - 162516009222144t^{16} - 55402309877760t^{14} - 9685466873856t^{12} - 865661091840t^{10} - 39676760064t^{8} - 951810048t^{6} - 11449728t^{4} - 56160t^{2} - 27\] \[B(t) = -972777519512027136t^{36} + 57637068031087607808t^{34} + 461689330549653504000t^{32} + 1353346324723623002112t^{30} + 2097052787973449318400t^{28} + 1950740731684831887360t^{26} + 1148018767415186817024t^{24} + 434953987064309219328t^{22} + 106067792243504185344t^{20} + 16572776343763156992t^{18} + 1657309253804752896t^{16} + 106189938248122368t^{14} + 4379344052944896t^{12} + 116273208360960t^{10} + 1953032601600t^{8} + 19693780992t^{6} + 104976000t^{4} + 204768t^{2} - 54\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 128288352x + 559236806388$, with conductor $13872$
Generic density of odd order reductions $299/2688$

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