The modular curve $X_{66a}$

Curve name $X_{66a}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 6 \\ 6 & 11 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 10 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 2 & 11 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{24}$
$8$ $24$ $X_{66}$
Meaning/Special name
Chosen covering $X_{66}$
Curves that $X_{66a}$ minimally covers
Curves that minimally cover $X_{66a}$
Curves that minimally cover $X_{66a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -81t^{20} - 3240t^{19} - 61344t^{18} - 730944t^{17} - 6039360t^{16} - 35334144t^{15} - 140009472t^{14} - 298598400t^{13} + 301916160t^{12} + 5037686784t^{11} + 19782696960t^{10} + 40301494272t^{9} + 19322634240t^{8} - 152882380800t^{7} - 573478797312t^{6} - 1157829230592t^{5} - 1583181987840t^{4} - 1532900671488t^{3} - 1029181538304t^{2} - 434865438720t - 86973087744\] \[B(t) = 3888t^{29} + 225504t^{28} + 6099840t^{27} + 102083328t^{26} + 1179712512t^{25} + 9913466880t^{24} + 61686448128t^{23} + 279105896448t^{22} + 825962987520t^{21} + 717177618432t^{20} - 7840360562688t^{19} - 52890933657600t^{18} - 184839894466560t^{17} - 363444226228224t^{16} + 2907553809825792t^{14} + 11829753245859840t^{13} + 27080158032691200t^{12} + 32114116864770048t^{11} - 23500476200779776t^{10} - 216521241400442880t^{9} - 585327488947716096t^{8} - 1034926864516251648t^{7} - 1330563001236848640t^{6} - 1266706664430501888t^{5} - 876889110453682176t^{4} - 419177812973322240t^{3} - 123972135054999552t^{2} - 17099604835172352t\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1000825x - 384846000$, with conductor $39200$
Generic density of odd order reductions $9249/57344$

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