## The modular curve $X_{212h}$

Curve name $X_{212h}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $48$ $X_{86j}$
Meaning/Special name
Chosen covering $X_{212}$
Curves that $X_{212h}$ minimally covers
Curves that minimally cover $X_{212h}$
Curves that minimally cover $X_{212h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -452984832t^{16} + 6794772480t^{14} - 3822059520t^{12} + 743178240t^{10} - 120987648t^{8} + 11612160t^{6} - 933120t^{4} + 25920t^{2} - 27$ $B(t) = 3710851743744t^{24} + 116891829927936t^{22} - 241089399226368t^{20} + 111412525400064t^{18} - 30166071902208t^{16} + 5536380616704t^{14} - 824036032512t^{12} + 86505947136t^{10} - 7364763648t^{8} + 425005056t^{6} - 14370048t^{4} + 108864t^{2} + 54$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 110x - 880$, with conductor $15$
Generic density of odd order reductions $19/336$