The modular curve $X_{211b}$

Curve name $X_{211b}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 24 & 7 \end{matrix}\right], \left[ \begin{matrix} 11 & 11 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 9 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 11 & 11 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{85}$
$16$ $48$ $X_{211}$
Meaning/Special name
Chosen covering $X_{211}$
Curves that $X_{211b}$ minimally covers
Curves that minimally cover $X_{211b}$
Curves that minimally cover $X_{211b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{26} - 12744t^{25} - 129168t^{24} + 564192t^{23} + 3427488t^{22} + 6023808t^{21} - 26977536t^{20} - 71788032t^{19} - 486487296t^{18} - 803229696t^{17} + 436396032t^{16} - 1836269568t^{15} + 13586448384t^{14} + 7345078272t^{13} + 6982336512t^{12} + 51406700544t^{11} - 124540747776t^{10} + 73510944768t^{9} - 110499987456t^{8} - 98694070272t^{7} + 224623853568t^{6} - 147899547648t^{5} - 135442464768t^{4} + 53452210176t^{3} - 452984832t^{2}\] \[B(t) = -54t^{39} + 55080t^{38} + 2939328t^{37} + 10208160t^{36} - 125776800t^{35} - 909066240t^{34} + 1368299520t^{33} + 9254559744t^{32} + 44472748032t^{31} + 86997639168t^{30} - 280366645248t^{29} - 1243674279936t^{28} - 4936181907456t^{27} - 11964277850112t^{26} + 4176364437504t^{25} - 6929648713728t^{24} + 154047068504064t^{23} + 271474517606400t^{22} + 1085898070425600t^{20} - 2464753096065024t^{19} - 443497517678592t^{18} - 1069149296001024t^{17} - 12251420518514688t^{16} + 20218601092939776t^{15} - 20376359402471424t^{14} + 18374108462972928t^{13} + 22805909122056192t^{12} - 46633056240402432t^{11} + 38816436952498176t^{10} - 22956256599736320t^{9} - 61006402667151360t^{8} + 33762952662220800t^{7} + 10960928338083840t^{6} - 12624317632217088t^{5} + 946267194654720t^{4} + 3710851743744t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 48020000x - 128100018750$, with conductor $1050$
Generic density of odd order reductions $1091/10752$

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