| 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 |
|
| 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 |
$9249/57344$ |