The modular curve $X_{215k}$

Curve name $X_{215k}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 14 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{25}$
$8$ $48$ $X_{96s}$
Meaning/Special name
Chosen covering $X_{215}$
Curves that $X_{215k}$ minimally covers
Curves that minimally cover $X_{215k}$
Curves that minimally cover $X_{215k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{24} - 216t^{20} + 3456t^{12} - 55296t^{4} - 110592\] \[B(t) = -54t^{36} - 648t^{32} - 1296t^{28} + 12096t^{24} + 82944t^{20} + 331776t^{16} + 774144t^{12} - 1327104t^{8} - 10616832t^{4} - 14155776\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 2008x - 33488$, with conductor $1200$
Generic density of odd order reductions $5/42$

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