The modular curve $X_{66e}$

Curve name $X_{66e}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $24$ $X_{24e}$
Meaning/Special name
Chosen covering $X_{66}$
Curves that $X_{66e}$ minimally covers
Curves that minimally cover $X_{66e}$
Curves that minimally cover $X_{66e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -81t^{8} - 1296t^{7} - 12096t^{6} - 72576t^{5} - 266112t^{4} - 580608t^{3} - 774144t^{2} - 663552t - 331776\] \[B(t) = 3888t^{11} + 85536t^{10} + 874368t^{9} + 5474304t^{8} + 23721984t^{7} + 76640256t^{6} + 189775872t^{5} + 350355456t^{4} + 447676416t^{3} + 350355456t^{2} + 127401984t\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 817x + 8976$, with conductor $1120$
Generic density of odd order reductions $307/2688$

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