## The modular curve $X_{37c}$

Curve name $X_{37c}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 10 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{9}$ $8$ $12$ $X_{37}$
Meaning/Special name
Chosen covering $X_{37}$
Curves that $X_{37c}$ minimally covers
Curves that minimally cover $X_{37c}$
Curves that minimally cover $X_{37c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -49545216t^{12} - 191102976t^{11} - 332660736t^{10} - 353009664t^{9} - 258453504t^{8} - 138682368t^{7} - 56180736t^{6} - 17335296t^{5} - 4038336t^{4} - 689472t^{3} - 81216t^{2} - 5832t - 189$ $B(t) = -123211874304t^{18} - 684913065984t^{17} - 1747615481856t^{16} - 2723344809984t^{15} - 2897970462720t^{14} - 2216794521600t^{13} - 1236960018432t^{12} - 488968814592t^{11} - 117252292608t^{10} + 14656536576t^{8} + 7640137728t^{7} + 2415937536t^{6} + 541209600t^{5} + 88439040t^{4} + 10388736t^{3} + 833328t^{2} + 40824t + 918$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 6633x + 210137$, with conductor $3200$
Generic density of odd order reductions $85091/344064$