The modular curve $X_{227b}$

Curve name $X_{227b}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 5 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{84}$
$16$ $48$ $X_{227}$
Meaning/Special name
Chosen covering $X_{227}$
Curves that $X_{227b}$ minimally covers
Curves that minimally cover $X_{227b}$
Curves that minimally cover $X_{227b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1811939328t^{24} + 54358179840t^{22} - 60926459904t^{20} + 16986931200t^{18} - 106168320t^{16} - 1274019840t^{14} + 490143744t^{12} - 79626240t^{10} - 414720t^{8} + 4147200t^{6} - 929664t^{4} + 51840t^{2} - 108\] \[B(t) = 29686813949952t^{36} + 1870269278846976t^{34} - 7720427052859392t^{32} + 6779726135820288t^{30} - 2414372915773440t^{28} + 102280351186944t^{26} + 211663504539648t^{24} - 94061394395136t^{22} + 20658826248192t^{20} - 1291176640512t^{16} + 367427321856t^{14} - 51675660288t^{12} - 1560674304t^{10} + 2302525440t^{8} - 404103168t^{6} + 28760832t^{4} - 435456t^{2} - 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 37359967x - 28559579937$, with conductor $196800$
Generic density of odd order reductions $299/2688$

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