| Curve name |
$X_{207k}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207k}$ minimally covers |
|
| Curves that minimally cover $X_{207k}$ |
|
| Curves that minimally cover $X_{207k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} - 1296t^{23} - 1728t^{22} + 798336t^{21} + 18493056t^{20} +
194724864t^{19} + 997429248t^{18} + 644308992t^{17} - 22557118464t^{16} -
127908052992t^{15} - 194401271808t^{14} + 879102001152t^{13} +
4824996249600t^{12} + 7032816009216t^{11} - 12441681395712t^{10} -
65488923131904t^{9} - 92393957228544t^{8} + 21112717049856t^{7} +
261470092787712t^{6} + 408367637987328t^{5} + 310261995012096t^{4} +
107150844100608t^{3} - 1855425871872t^{2} - 11132555231232t - 1855425871872\]
\[B(t) = 54t^{36} + 3888t^{35} + 238464t^{34} + 10046592t^{33} + 254399616t^{32}
+ 3873153024t^{31} + 33095761920t^{30} + 75687395328t^{29} - 1707510398976t^{28}
- 22892480299008t^{27} - 133631346475008t^{26} - 274915815063552t^{25} +
1573892627890176t^{24} + 14008766114562048t^{23} + 39431776984104960t^{22} -
36599920563585024t^{21} - 631426731554635776t^{20} - 1815998115581263872t^{19} +
14527984924650110976t^{17} + 40411310819496689664t^{16} +
18739159328555532288t^{15} - 161512558526893916160t^{14} -
459039248041969188864t^{13} - 412586509045642297344t^{12} +
576540251392158203904t^{11} + 2241961964182047817728t^{10} +
3072576694017614413824t^{9} + 1833425330295457972224t^{8} -
650149775306366386176t^{7} - 2274323441321634693120t^{6} -
2129288393021888397312t^{5} - 1118861343574997336064t^{4} -
353483031152682860544t^{3} - 67121648846329872384t^{2} - 8754997675608244224t -
972777519512027136\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 + 62229x + 270705693$, with conductor $1470$ |
| Generic density of odd order reductions |
$11/112$ |