The modular curve $X_{117a}$

Curve name $X_{117a}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{36e}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117a}$ minimally covers
Curves that minimally cover $X_{117a}$
Curves that minimally cover $X_{117a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1728t^{16} + 7776t^{12} - 5292t^{8} + 1296t^{4} - 108\] \[B(t) = -27648t^{24} - 186624t^{20} + 316224t^{16} - 190512t^{12} + 55080t^{8} - 7776t^{4} + 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 12909409x - 324219616895$, with conductor $1138368$
Generic density of odd order reductions $335/2688$

Back to the 2-adic image homepage.