## The modular curve $X_{225i}$

Curve name $X_{225i}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $48$ $X_{85n}$
Meaning/Special name
Chosen covering $X_{225}$
Curves that $X_{225i}$ minimally covers
Curves that minimally cover $X_{225i}$
Curves that minimally cover $X_{225i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -7421703487488t^{24} - 111325552312320t^{22} - 62852551409664t^{20} - 15655155793920t^{18} - 3940968038400t^{16} - 597939978240t^{14} - 92522151936t^{12} - 9342812160t^{10} - 962150400t^{8} - 59719680t^{6} - 3746304t^{4} - 103680t^{2} - 108$ $B(t) = -7782220156096217088t^{36} + 245139934917030838272t^{34} + 505236324196559093760t^{32} + 245139934917030838272t^{30} + 86957190455129800704t^{28} + 22742474430779228160t^{26} + 5063857976327012352t^{24} + 897729253846548480t^{22} + 144714868589592576t^{20} + 18790361660915712t^{18} + 2261169821712384t^{16} + 219172181114880t^{14} + 19317085175808t^{12} + 1355557109760t^{10} + 80985194496t^{8} + 3567255552t^{6} + 114877440t^{4} + 870912t^{2} - 432$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 3456033x + 2471796063$, with conductor $4800$
Generic density of odd order reductions $271/2688$