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

Curve name $X_{101g}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
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 $X_{101g}$ minimally covers
Curves that minimally cover $X_{101g}$
Curves that minimally cover $X_{101g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -442368t^{8} + 221184t^{6} - 34560t^{4} + 1728t^{2} - 108$ $B(t) = -113246208t^{12} + 84934656t^{10} - 23887872t^{8} + 3096576t^{6} - 124416t^{4} - 10368t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 3137x - 66495$, with conductor $1344$
Generic density of odd order reductions $635/5376$