The modular curve $X_{123c}$

Curve name $X_{123c}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 15 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{9}$
$8$ $24$ $X_{37d}$
Meaning/Special name
Chosen covering $X_{123}$
Curves that $X_{123c}$ minimally covers
Curves that minimally cover $X_{123c}$
Curves that minimally cover $X_{123c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -756t^{16} - 17280t^{15} - 167616t^{14} - 940032t^{13} - 3571776t^{12} - 10188288t^{11} - 22844160t^{10} - 39426048t^{9} - 48567168t^{8} - 36882432t^{7} - 9262080t^{6} + 11059200t^{5} + 11860992t^{4} + 3760128t^{3} - 774144t^{2} - 884736t - 193536\] \[B(t) = 7344t^{24} + 238464t^{23} + 3338496t^{22} + 26051328t^{21} + 118081152t^{20} + 241325568t^{19} - 574387200t^{18} - 6514172928t^{17} - 27112962816t^{16} - 72876478464t^{15} - 139910602752t^{14} - 196666859520t^{13} - 198831919104t^{12} - 130997108736t^{11} - 29825335296t^{10} + 44538494976t^{9} + 60002684928t^{8} + 33005076480t^{7} + 2574581760t^{6} - 8668643328t^{5} - 5443780608t^{4} - 803340288t^{3} + 488374272t^{2} + 233570304t + 30081024\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 304649x - 64823033$, with conductor $1664$
Generic density of odd order reductions $12833/57344$

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