## The modular curve $X_{235j}$

Curve name $X_{235j}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{102}$ $16$ $48$ $X_{235}$
Meaning/Special name
Chosen covering $X_{235}$
Curves that $X_{235j}$ minimally covers
Curves that minimally cover $X_{235j}$
Curves that minimally cover $X_{235j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{22} + 270t^{20} - 783t^{18} + 648t^{16} + 378t^{14} - 972t^{12} + 378t^{10} + 648t^{8} - 783t^{6} + 270t^{4} - 27t^{2}$ $B(t) = 54t^{33} - 810t^{31} + 4374t^{29} - 10314t^{27} + 8910t^{25} + 6318t^{23} - 17874t^{21} + 14094t^{19} - 14094t^{17} + 17874t^{15} - 6318t^{13} - 8910t^{11} + 10314t^{9} - 4374t^{7} + 810t^{5} - 54t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 120948x - 12199696$, with conductor $20160$
Generic density of odd order reductions $11/112$