The modular curve $X_{108b}$

Curve name $X_{108b}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 10 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 10 & 5 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{10d}$
$8$ $24$ $X_{42d}$
Meaning/Special name
Chosen covering $X_{108}$
Curves that $X_{108b}$ minimally covers
Curves that minimally cover $X_{108b}$
Curves that minimally cover $X_{108b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = 54t^{16} - 432t^{15} - 5616t^{14} + 864t^{13} + 132192t^{12} + 319680t^{11} - 630720t^{10} - 3874176t^{9} - 5358528t^{8} + 2744064t^{7} + 18005760t^{6} + 25698816t^{5} + 19367424t^{4} + 8487936t^{3} + 2128896t^{2} + 276480t + 13824\] \[B(t) = -540t^{24} - 5184t^{23} + 12960t^{22} + 262656t^{21} + 308448t^{20} - 4997376t^{19} - 15189120t^{18} + 39813120t^{17} + 247209408t^{16} + 86648832t^{15} - 1871133696t^{14} - 4676050944t^{13} + 223921152t^{12} + 23439974400t^{11} + 58742599680t^{10} + 79136980992t^{9} + 65681611776t^{8} + 31456346112t^{7} + 3944816640t^{6} - 5584453632t^{5} - 4380106752t^{4} - 1633222656t^{3} - 353009664t^{2} - 42467328t - 2211840\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 19039x + 1017759$, with conductor $1664$
Generic density of odd order reductions $12833/57344$

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