The modular curve $X_{211f}$

Curve name $X_{211f}$
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} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 11 & 11 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 9 & 0 \\ 16 & 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 $X_{211f}$ minimally covers
Curves that minimally cover $X_{211f}$
Curves that minimally cover $X_{211f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{26} - 50976t^{25} - 516672t^{24} + 2256768t^{23} + 13709952t^{22} + 24095232t^{21} - 107910144t^{20} - 287152128t^{19} - 1945949184t^{18} - 3212918784t^{17} + 1745584128t^{16} - 7345078272t^{15} + 54345793536t^{14} + 29380313088t^{13} + 27929346048t^{12} + 205626802176t^{11} - 498162991104t^{10} + 294043779072t^{9} - 441999949824t^{8} - 394776281088t^{7} + 898495414272t^{6} - 591598190592t^{5} - 541769859072t^{4} + 213808840704t^{3} - 1811939328t^{2}\] \[B(t) = -432t^{39} + 440640t^{38} + 23514624t^{37} + 81665280t^{36} - 1006214400t^{35} - 7272529920t^{34} + 10946396160t^{33} + 74036477952t^{32} + 355781984256t^{31} + 695981113344t^{30} - 2242933161984t^{29} - 9949394239488t^{28} - 39489455259648t^{27} - 95714222800896t^{26} + 33410915500032t^{25} - 55437189709824t^{24} + 1232376548032512t^{23} + 2171796140851200t^{22} + 8687184563404800t^{20} - 19718024768520192t^{19} - 3547980141428736t^{18} - 8553194368008192t^{17} - 98011364148117504t^{16} + 161748808743518208t^{15} - 163010875219771392t^{14} + 146992867703783424t^{13} + 182447272976449536t^{12} - 373064449923219456t^{11} + 310531495619985408t^{10} - 183650052797890560t^{9} - 488051221337210880t^{8} + 270103621297766400t^{7} + 87687426704670720t^{6} - 100994541057736704t^{5} + 7570137557237760t^{4} + 29686813949952t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 432180005x + 3458268326247$, with conductor $3150$
Generic density of odd order reductions $11/112$

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