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

Curve name $X_{75f}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{75f}$ minimally covers
Curves that minimally cover $X_{75f}$
Curves that minimally cover $X_{75f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -442368t^{8} + 3317760t^{6} - 926208t^{4} + 51840t^{2} - 108$ $B(t) = -113246208t^{12} - 1783627776t^{10} + 1842020352t^{8} - 390168576t^{6} + 28781568t^{4} - 435456t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 + 55743x + 1756863$, with conductor $14784$
Generic density of odd order reductions $307/2688$