## The modular curve $X_{101p}$

Curve name $X_{101p}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{101}$
Curves that $X_{101p}$ minimally covers
Curves that minimally cover $X_{101p}$
Curves that minimally cover $X_{101p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -7077888t^{12} + 7077888t^{10} - 2764800t^{8} + 525312t^{6} - 50112t^{4} + 2592t^{2} - 108$ $B(t) = -7247757312t^{18} + 10871635968t^{16} - 6964641792t^{14} + 2477260800t^{12} - 528187392t^{10} + 66355200t^{8} - 4064256t^{6} - 20736t^{4} + 15552t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 63720x - 5613300$, with conductor $2070$
Generic density of odd order reductions $193/1792$