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

Curve name $X_{101c}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
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 $X_{101c}$ minimally covers
Curves that minimally cover $X_{101c}$
Curves that minimally cover $X_{101c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -452984832t^{16} + 566231040t^{14} - 297271296t^{12} + 84934656t^{10} - 14376960t^{8} + 1492992t^{6} - 98496t^{4} + 4320t^{2} - 108$ $B(t) = -3710851743744t^{24} + 6957847019520t^{22} - 5827196878848t^{20} + 2873735774208t^{18} - 924089057280t^{16} + 201804742656t^{14} - 29974855680t^{12} + 2890432512t^{10} - 151953408t^{8} + 359424t^{6} + 476928t^{4} - 25920t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1383564x - 625494800$, with conductor $28224$
Generic density of odd order reductions $635/5376$