The modular curve $X_{229d}$

Curve name $X_{229d}$
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} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{86i}$
Meaning/Special name
Chosen covering $X_{229}$
Curves that $X_{229d}$ minimally covers
Curves that minimally cover $X_{229d}$
Curves that minimally cover $X_{229d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27648t^{16} - 110592t^{14} + 1575936t^{12} + 3345408t^{10} - 4544640t^{8} + 836352t^{6} + 98496t^{4} - 1728t^{2} - 108\] \[B(t) = 1769472t^{24} + 10616832t^{22} + 241532928t^{20} + 898007040t^{18} - 1176809472t^{16} - 3905667072t^{14} + 4125413376t^{12} - 976416768t^{10} - 73550592t^{8} + 14031360t^{6} + 943488t^{4} + 10368t^{2} + 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 14463x - 1157247$, with conductor $3264$
Generic density of odd order reductions $109/896$

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