The modular curve $X_{207k}$

Curve name $X_{207k}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{27}$
$8$ $48$ $X_{92i}$
Meaning/Special name
Chosen covering $X_{207}$
Curves that $X_{207k}$ minimally covers
Curves that minimally cover $X_{207k}$
Curves that minimally cover $X_{207k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{24} - 1296t^{23} - 1728t^{22} + 798336t^{21} + 18493056t^{20} + 194724864t^{19} + 997429248t^{18} + 644308992t^{17} - 22557118464t^{16} - 127908052992t^{15} - 194401271808t^{14} + 879102001152t^{13} + 4824996249600t^{12} + 7032816009216t^{11} - 12441681395712t^{10} - 65488923131904t^{9} - 92393957228544t^{8} + 21112717049856t^{7} + 261470092787712t^{6} + 408367637987328t^{5} + 310261995012096t^{4} + 107150844100608t^{3} - 1855425871872t^{2} - 11132555231232t - 1855425871872\] \[B(t) = 54t^{36} + 3888t^{35} + 238464t^{34} + 10046592t^{33} + 254399616t^{32} + 3873153024t^{31} + 33095761920t^{30} + 75687395328t^{29} - 1707510398976t^{28} - 22892480299008t^{27} - 133631346475008t^{26} - 274915815063552t^{25} + 1573892627890176t^{24} + 14008766114562048t^{23} + 39431776984104960t^{22} - 36599920563585024t^{21} - 631426731554635776t^{20} - 1815998115581263872t^{19} + 14527984924650110976t^{17} + 40411310819496689664t^{16} + 18739159328555532288t^{15} - 161512558526893916160t^{14} - 459039248041969188864t^{13} - 412586509045642297344t^{12} + 576540251392158203904t^{11} + 2241961964182047817728t^{10} + 3072576694017614413824t^{9} + 1833425330295457972224t^{8} - 650149775306366386176t^{7} - 2274323441321634693120t^{6} - 2129288393021888397312t^{5} - 1118861343574997336064t^{4} - 353483031152682860544t^{3} - 67121648846329872384t^{2} - 8754997675608244224t - 972777519512027136\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 + 62229x + 270705693$, with conductor $1470$
Generic density of odd order reductions $11/112$

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