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

Curve name $X_{75j}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{75}$
Curves that $X_{75j}$ minimally covers
Curves that minimally cover $X_{75j}$
Curves that minimally cover $X_{75j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -110592t^{8} + 829440t^{6} - 231552t^{4} + 12960t^{2} - 27$ $B(t) = -14155776t^{12} - 222953472t^{10} + 230252544t^{8} - 48771072t^{6} + 3597696t^{4} - 54432t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 + 13936x + 212640$, with conductor $3696$
Generic density of odd order reductions $65/896$