The modular curve $X_{212b}$

Curve name $X_{212b}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{86}$
$16$ $48$ $X_{212}$
Meaning/Special name
Chosen covering $X_{212}$
Curves that $X_{212b}$ minimally covers
Curves that minimally cover $X_{212b}$
Curves that minimally cover $X_{212b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -7599824371187712t^{32} + 113997365567815680t^{30} - 64123518131896320t^{28} + 12468461858979840t^{26} - 2026125052084224t^{24} + 139156940390400t^{22} + 15655155793920t^{20} - 5653250703360t^{18} + 990224842752t^{16} - 88332042240t^{14} + 3822059520t^{12} + 530841600t^{10} - 120766464t^{8} + 11612160t^{6} - 933120t^{4} + 25920t^{2} - 27\] \[B(t) = 255007790074960841539584t^{48} + 8032745387361266508496896t^{46} - 16567537361432612173774848t^{44} + 7656210447328707140911104t^{42} - 2073183449584032232243200t^{40} + 374573820553223120879616t^{38} - 44492898187441097146368t^{36} + 337067410510917402624t^{34} + 1012251007295976112128t^{32} - 248012668529339793408t^{30} + 37525082810785726464t^{28} - 2977468691924385792t^{26} + 46522948311318528t^{22} - 9161397170601984t^{20} + 946093248479232t^{18} - 60334861713408t^{16} - 313918488576t^{14} + 647456882688t^{12} - 85168226304t^{10} + 7365427200t^{8} - 425005056t^{6} + 14370048t^{4} - 108864t^{2} - 54\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 24755x + 2820872$, with conductor $225$
Generic density of odd order reductions $299/2688$

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