| Curve name |
$X_{207j}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207j}$ minimally covers |
|
| Curves that minimally cover $X_{207j}$ |
|
| Curves that minimally cover $X_{207j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 3456t^{15} + 55296t^{14} + 2515968t^{13} + 29611008t^{12}
+ 147087360t^{11} + 123863040t^{10} - 2057895936t^{9} - 9911697408t^{8} -
16463167488t^{7} + 7927234560t^{6} + 75308728320t^{5} + 121286688768t^{4} +
82443239424t^{3} + 14495514624t^{2} - 7247757312t - 1811939328\]
\[B(t) = 432t^{24} + 20736t^{23} + 1327104t^{22} + 44402688t^{21} +
709502976t^{20} + 5081481216t^{19} - 2972712960t^{18} - 360579465216t^{17} -
3066284408832t^{16} - 12049906139136t^{15} - 13122064613376t^{14} +
91578584530944t^{13} + 457518757183488t^{12} + 732628676247552t^{11} -
839812135256064t^{10} - 6169551943237632t^{9} - 12559500938575872t^{8} -
11815467916197888t^{7} - 779278866186240t^{6} + 10656638495096832t^{5} +
11903484680994816t^{4} + 5959627900452864t^{3} + 1424967069597696t^{2} +
178120883699712t + 29686813949952\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 81279x - 404073855$, with conductor $6720$ |
| Generic density of odd order reductions |
$299/2688$ |