The modular curve $X_{78j}$

Curve name	$X_{78j}$
Index	$48$
Level	$8$
Genus	$0$
Does the subgroup contain $-I$?	No
Generating matrices	$\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels	Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering	$X_{78}$
Curves that $X_{78j}$ minimally covers
Curves that minimally cover $X_{78j}$
Curves that minimally cover $X_{78j}$ and have infinitely many rational points.
Model	$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $$y^2 = x^3 + A(t)x + B(t), \text{ where}$$ $$A(t) = -27t^{8} - 3240t^{6} - 14472t^{4} - 12960t^{2} - 432$$ $$B(t) = -54t^{12} + 13608t^{10} + 224856t^{8} + 762048t^{6} + 899424t^{4} + 217728t^{2} - 3456$$
Info about rational points
Comments on finding rational points	None
Elliptic curve whose $2$-adic image is the subgroup	$y^2 = x^3 - x^2 - 47264x + 3967008$, with conductor $3696$
Generic density of odd order reductions	$65/896$

Back to the 2-adic image homepage.