The modular curve $X_{119j}$

Curve name $X_{119j}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{36s}$
Meaning/Special name
Chosen covering $X_{119}$
Curves that $X_{119j}$ minimally covers
Curves that minimally cover $X_{119j}$
Curves that minimally cover $X_{119j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{12} - 648t^{10} - 6048t^{8} - 27648t^{6} - 62640t^{4} - 58752t^{2} - 6912\] \[B(t) = -54t^{18} - 1944t^{16} - 29808t^{14} - 254016t^{12} - 1312848t^{10} - 4193856t^{8} - 8007552t^{6} - 8169984t^{4} - 3151872t^{2} + 221184\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 80008x - 8683988$, with conductor $600$
Generic density of odd order reductions $41/336$

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