The modular curve $X_{219e}$

Curve name $X_{219e}$
Index $96$
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} 5 & 5 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $48$ $X_{85n}$
Meaning/Special name
Chosen covering $X_{219}$
Curves that $X_{219e}$ minimally covers
Curves that minimally cover $X_{219e}$
Curves that minimally cover $X_{219e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -187596t^{24} + 18144t^{23} - 397008t^{22} - 2850336t^{21} - 565272t^{20} - 7299936t^{19} - 17609616t^{18} - 1956960t^{17} - 78137460t^{16} + 26847936t^{15} - 181169568t^{14} + 24373440t^{13} - 212260176t^{12} - 24373440t^{11} - 181169568t^{10} - 26847936t^{9} - 78137460t^{8} + 1956960t^{7} - 17609616t^{6} + 7299936t^{5} - 565272t^{4} + 2850336t^{3} - 397008t^{2} - 18144t - 187596\] \[B(t) = 31271184t^{36} - 4432320t^{35} + 97565472t^{34} + 727565760t^{33} + 13552272t^{32} + 3365079552t^{31} + 3983254272t^{30} + 12763381248t^{29} + 13853954880t^{28} + 50790212352t^{27} + 72542646144t^{26} + 54415348992t^{25} + 384962849856t^{24} - 99036504576t^{23} + 1011424548096t^{22} - 228023907840t^{21} + 1650526686432t^{20} - 144222996096t^{19} + 1922680601280t^{18} + 144222996096t^{17} + 1650526686432t^{16} + 228023907840t^{15} + 1011424548096t^{14} + 99036504576t^{13} + 384962849856t^{12} - 54415348992t^{11} + 72542646144t^{10} - 50790212352t^{9} + 13853954880t^{8} - 12763381248t^{7} + 3983254272t^{6} - 3365079552t^{5} + 13552272t^{4} - 727565760t^{3} + 97565472t^{2} + 4432320t + 31271184\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 2316x + 42896$, with conductor $576$
Generic density of odd order reductions $109/896$

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