The modular curve $X_{66b}$

Curve name $X_{66b}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 6 \\ 2 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{24}$
Meaning/Special name
Chosen covering $X_{66}$
Curves that $X_{66b}$ minimally covers
Curves that minimally cover $X_{66b}$
Curves that minimally cover $X_{66b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -81t^{16} - 2592t^{15} - 38016t^{14} - 338688t^{13} - 1942272t^{12} - 6359040t^{11} - 2875392t^{10} + 73875456t^{9} + 334209024t^{8} + 591003648t^{7} - 184025088t^{6} - 3255828480t^{5} - 7955546112t^{4} - 11098128384t^{3} - 9965666304t^{2} - 5435817984t - 1358954496\] \[B(t) = 3888t^{23} + 178848t^{22} + 3673728t^{21} + 44126208t^{20} + 337665024t^{19} + 1637922816t^{18} + 4167770112t^{17} - 3609722880t^{16} - 80372957184t^{15} - 350758895616t^{14} - 751756640256t^{13} + 6014053122048t^{11} + 22448569319424t^{10} + 41150954078208t^{9} + 14785424916480t^{8} - 136569491030016t^{7} - 429371638677504t^{6} - 708134880411648t^{5} - 740314922876928t^{4} - 493079425449984t^{3} - 192036577738752t^{2} - 33397665693696t\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 40033x - 3078768$, with conductor $7840$
Generic density of odd order reductions $149/896$

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