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

Curve name $X_{101i}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
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 $X_{101i}$ minimally covers
Curves that minimally cover $X_{101i}$
Curves that minimally cover $X_{101i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -28311552t^{12} + 21233664t^{10} - 6193152t^{8} + 884736t^{6} - 69120t^{4} + 3456t^{2} - 108$ $B(t) = 57982058496t^{18} - 65229815808t^{16} + 31255953408t^{14} - 8323596288t^{12} + 1316487168t^{10} - 116785152t^{8} + 3870720t^{6} + 207360t^{4} - 20736t^{2} + 432$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 3431x + 59961$, with conductor $1470$
Generic density of odd order reductions $193/1792$