The modular curve $X_{85g}$

Curve name $X_{85g}$
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} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{85}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{85g}$ minimally covers
Curves that minimally cover $X_{85g}$
Curves that minimally cover $X_{85g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{16} + 2160t^{14} - 11664t^{12} - 20736t^{10} + 423360t^{8} - 1596672t^{6} + 2509056t^{4} - 1382400t^{2} - 110592\] \[B(t) = 432t^{24} - 12960t^{22} + 228096t^{20} - 2785536t^{18} + 22726656t^{16} - 120434688t^{14} + 406038528t^{12} - 827117568t^{10} + 856313856t^{8} - 58613760t^{6} - 729907200t^{4} + 498991104t^{2} - 14155776\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 240051x + 126936754$, with conductor $7056$
Generic density of odd order reductions $41/336$

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