The modular curve $X_{116f}$

Curve name $X_{116f}$
Index $48$
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} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13f}$
$8$ $24$ $X_{32f}$
Meaning/Special name
Chosen covering $X_{116}$
Curves that $X_{116f}$ minimally covers
Curves that minimally cover $X_{116f}$
Curves that minimally cover $X_{116f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{16} - 1296t^{12} - 21168t^{8} - 124416t^{4} - 110592\] \[B(t) = -54t^{24} - 3888t^{20} - 110160t^{16} - 1524096t^{12} - 10119168t^{8} - 23887872t^{4} + 14155776\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 6153682x - 5147588002$, with conductor $28227$
Generic density of odd order reductions $419/2688$

Back to the 2-adic image homepage.