## The modular curve $X_{75h}$

Curve name $X_{75h}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{75h}$ minimally covers
Curves that minimally cover $X_{75h}$
Curves that minimally cover $X_{75h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -7077888t^{12} + 54853632t^{10} - 28200960t^{8} + 5363712t^{6} - 440640t^{4} + 13392t^{2} - 27$ $B(t) = -7247757312t^{18} - 111434268672t^{16} + 160356630528t^{14} - 74516004864t^{12} + 16955080704t^{10} - 2119385088t^{8} + 145539072t^{6} - 4893696t^{4} + 53136t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 682848x - 74301228$, with conductor $25872$
Generic density of odd order reductions $193/1792$