The modular curve $X_{84b}$

Curve name $X_{84b}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 3 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{84}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{84b}$ minimally covers
Curves that minimally cover $X_{84b}$
Curves that minimally cover $X_{84b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{12} - 648t^{10} + 108t^{8} + 4752t^{6} + 7020t^{4} + 2808t^{2} - 108\] \[B(t) = 432t^{18} + 3888t^{16} + 28512t^{14} + 127008t^{12} + 285120t^{10} + 300672t^{8} + 102816t^{6} - 44064t^{4} - 29808t^{2} - 432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 8033x + 2375937$, with conductor $4800$
Generic density of odd order reductions $691/5376$

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