## The modular curve $X_{121h}$

Curve name $X_{121h}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36e}$
Meaning/Special name
Chosen covering $X_{121}$
Curves that $X_{121h}$ minimally covers
Curves that minimally cover $X_{121h}$
Curves that minimally cover $X_{121h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 1728t^{10} - 10800t^{8} - 32832t^{6} - 48492t^{4} - 28512t^{2} - 1728$ $B(t) = 432t^{18} + 10368t^{16} + 106272t^{14} + 604800t^{12} + 2076840t^{10} + 4364928t^{8} + 5397840t^{6} + 3478464t^{4} + 808704t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 1533x + 22563$, with conductor $4800$
Generic density of odd order reductions $635/5376$