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

Curve name $X_{101f}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 1 \end{matrix}\right]$
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 $X_{101f}$ minimally covers
Curves that minimally cover $X_{101f}$
Curves that minimally cover $X_{101f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -113246208t^{16} + 141557760t^{14} - 74317824t^{12} + 21233664t^{10} - 3594240t^{8} + 373248t^{6} - 24624t^{4} + 1080t^{2} - 27$ $B(t) = 463856467968t^{24} - 869730877440t^{22} + 728399609856t^{20} - 359216971776t^{18} + 115511132160t^{16} - 25225592832t^{14} + 3746856960t^{12} - 361304064t^{10} + 18994176t^{8} - 44928t^{6} - 59616t^{4} + 3240t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 345891x + 78186850$, with conductor $7056$
Generic density of odd order reductions $41/336$