The modular curve $X_{82c}$

Curve name $X_{82c}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 5 & 15 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 14 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 14 & 7 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{11}$
$8$ $24$ $X_{82}$
Meaning/Special name
Chosen covering $X_{82}$
Curves that $X_{82c}$ minimally covers
Curves that minimally cover $X_{82c}$
Curves that minimally cover $X_{82c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -1890t^{12} - 24624t^{11} - 137592t^{10} - 453600t^{9} - 1046520t^{8} - 1897344t^{7} - 2894400t^{6} - 3794688t^{5} - 4186080t^{4} - 3628800t^{3} - 2201472t^{2} - 787968t - 120960\] \[B(t) = 31536t^{18} + 614304t^{17} + 5393952t^{16} + 28646784t^{15} + 104302080t^{14} + 278256384t^{13} + 558157824t^{12} + 820772352t^{11} + 755910144t^{10} - 1511820288t^{8} - 3283089408t^{7} - 4465262592t^{6} - 4452102144t^{5} - 3337666560t^{4} - 1833394176t^{3} - 690425856t^{2} - 157261824t - 16146432\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 23x + 51$, with conductor $768$
Generic density of odd order reductions $417/1792$

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