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

Curve name $X_{235f}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{102k}$
Meaning/Special name
Chosen covering $X_{235}$
Curves that $X_{235f}$ minimally covers
Curves that minimally cover $X_{235f}$
Curves that minimally cover $X_{235f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{24} + 540t^{22} - 3942t^{20} + 12204t^{18} - 12069t^{16} - 7560t^{14} + 14796t^{12} - 7560t^{10} - 12069t^{8} + 12204t^{6} - 3942t^{4} + 540t^{2} - 27$ $B(t) = 54t^{36} - 1620t^{34} + 19926t^{32} - 127872t^{30} + 445176t^{28} - 752976t^{26} + 234360t^{24} + 974592t^{22} - 1197180t^{20} + 589896t^{18} - 1197180t^{16} + 974592t^{14} + 234360t^{12} - 752976t^{10} + 445176t^{8} - 127872t^{6} + 19926t^{4} - 1620t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 + 10289x - 298411$, with conductor $1470$
Generic density of odd order reductions $11/112$