The modular curve $X_{120e}$

Curve name $X_{120e}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 3 \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 $X_{120}$
Curves that $X_{120e}$ minimally covers
Curves that minimally cover $X_{120e}$
Curves that minimally cover $X_{120e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{12} + 3456t^{10} - 43200t^{8} + 262656t^{6} - 775872t^{4} + 912384t^{2} - 110592\] \[B(t) = 432t^{18} - 20736t^{16} + 425088t^{14} - 4838400t^{12} + 33229440t^{10} - 139677696t^{8} + 345461760t^{6} - 445243392t^{4} + 207028224t^{2} + 14155776\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 823201x + 287410955$, with conductor $1470$
Generic density of odd order reductions $193/1792$

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