The modular curve $X_{79h}$

Curve name $X_{79h}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{79}$
Curves that $X_{79h}$ minimally covers
Curves that minimally cover $X_{79h}$
Curves that minimally cover $X_{79h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -891t^{8} - 3888t^{7} - 6264t^{6} - 2592t^{5} + 2808t^{4} - 5184t^{3} - 25056t^{2} - 31104t - 14256\] \[B(t) = 10206t^{12} + 66096t^{11} + 169128t^{10} + 135648t^{9} - 382968t^{8} - 1503360t^{7} - 2721600t^{6} - 3006720t^{5} - 1531872t^{4} + 1085184t^{3} + 2706048t^{2} + 2115072t + 653184\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 13067x + 574926$, with conductor $1120$
Generic density of odd order reductions $307/2688$

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