## The modular curve $X_{102k}$

Curve name $X_{102k}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{102}$
Curves that $X_{102k}$ minimally covers
Curves that minimally cover $X_{102k}$
Curves that minimally cover $X_{102k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 216t^{11} + 216t^{10} + 3456t^{9} - 3888t^{8} - 19008t^{7} + 45792t^{6} - 19008t^{5} - 62640t^{4} + 120960t^{3} - 100224t^{2} + 41472t - 6912$ $B(t) = 54t^{18} + 648t^{17} + 648t^{16} - 14688t^{15} - 18144t^{14} + 181440t^{13} + 18144t^{12} - 1259712t^{11} + 1686096t^{10} + 2158272t^{9} - 8589888t^{8} + 12752640t^{7} - 16232832t^{6} + 21772800t^{5} - 23680512t^{4} + 17252352t^{3} - 7796736t^{2} + 1990656t - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 30592x + 2036852$, with conductor $2352$
Generic density of odd order reductions $17/168$