## The modular curve $X_{234c}$

Curve name $X_{234c}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $48$ $X_{78g}$
Meaning/Special name
Chosen covering $X_{234}$
Curves that $X_{234c}$ minimally covers
Curves that minimally cover $X_{234c}$
Curves that minimally cover $X_{234c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -124524t^{16} - 1655424t^{15} - 10219392t^{14} - 39965184t^{13} - 106106112t^{12} - 210954240t^{11} - 339462144t^{10} - 310321152t^{9} - 677873664t^{8} + 1241284608t^{7} - 5431394304t^{6} + 13501071360t^{5} - 27163164672t^{4} + 40924348416t^{3} - 41858629632t^{2} + 27122466816t - 8160804864$ $B(t) = 16913232t^{24} + 337271040t^{23} + 3202861824t^{22} + 19500106752t^{21} + 84816087552t^{20} + 280654295040t^{19} + 735358611456t^{18} + 1577113141248t^{17} + 2757619593216t^{16} + 4058967048192t^{15} + 5120147718144t^{14} + 4317271031808t^{13} + 10302184488960t^{12} - 17269084127232t^{11} + 81922363490304t^{10} - 259773891084288t^{9} + 705950615863296t^{8} - 1614963856637952t^{7} + 3012028872523776t^{6} - 4598239969935360t^{5} + 5558507113807872t^{4} - 5111835984396288t^{3} + 3358444039962624t^{2} - 1414617272156160t + 283756946522112$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 1537x + 23713$, with conductor $192$
Generic density of odd order reductions $109/896$