The modular curve $X_{79f}$

Curve name $X_{79f}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 3 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $6$ $X_{13}$
$8$ $24$ $X_{79}$
Meaning/Special name
Chosen covering $X_{79}$
Curves that $X_{79f}$ minimally covers
Curves that minimally cover $X_{79f}$
Curves that minimally cover $X_{79f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -891t^{16} - 11016t^{15} - 51624t^{14} - 100656t^{13} + 8208t^{12} + 384480t^{11} + 538272t^{10} - 278208t^{9} - 1364256t^{8} - 556416t^{7} + 2153088t^{6} + 3075840t^{5} + 131328t^{4} - 3220992t^{3} - 3303936t^{2} - 1410048t - 228096\] \[B(t) = 10206t^{24} + 188568t^{23} + 1452168t^{22} + 5746032t^{21} + 9924768t^{20} - 11205216t^{19} - 99946656t^{18} - 222201792t^{17} - 143589024t^{16} + 433057536t^{15} + 1337990400t^{14} + 1579378176t^{13} - 3158756352t^{11} - 5351961600t^{10} - 3464460288t^{9} + 2297424384t^{8} + 7110457344t^{7} + 6396585984t^{6} + 1434267648t^{5} - 2540740608t^{4} - 2941968384t^{3} - 1487020032t^{2} - 386187264t - 41803776\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 640283x - 197199618$, with conductor $7840$
Generic density of odd order reductions $149/896$

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