The modular curve $X_{123f}$

Curve name $X_{123f}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 15 & 15 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{9}$
$8$ $24$ $X_{37b}$
Meaning/Special name
Chosen covering $X_{123}$
Curves that $X_{123f}$ minimally covers
Curves that minimally cover $X_{123f}$
Curves that minimally cover $X_{123f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -189t^{16} - 4320t^{15} - 41904t^{14} - 235008t^{13} - 892944t^{12} - 2547072t^{11} - 5711040t^{10} - 9856512t^{9} - 12141792t^{8} - 9220608t^{7} - 2315520t^{6} + 2764800t^{5} + 2965248t^{4} + 940032t^{3} - 193536t^{2} - 221184t - 48384\] \[B(t) = -918t^{24} - 29808t^{23} - 417312t^{22} - 3256416t^{21} - 14760144t^{20} - 30165696t^{19} + 71798400t^{18} + 814271616t^{17} + 3389120352t^{16} + 9109559808t^{15} + 17488825344t^{14} + 24583357440t^{13} + 24853989888t^{12} + 16374638592t^{11} + 3728166912t^{10} - 5567311872t^{9} - 7500335616t^{8} - 4125634560t^{7} - 321822720t^{6} + 1083580416t^{5} + 680472576t^{4} + 100417536t^{3} - 61046784t^{2} - 29196288t - 3760128\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 76162x + 8064798$, with conductor $1664$
Generic density of odd order reductions $45667/172032$

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