| Curve name |
$X_{226e}$ |
| 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} 9 & 9 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{226}$ |
| Curves that $X_{226e}$ minimally covers |
|
| Curves that minimally cover $X_{226e}$ |
|
| Curves that minimally cover $X_{226e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -3564t^{16} - 31104t^{15} - 74304t^{14} + 152064t^{13} + 1069632t^{12}
+ 428544t^{11} - 13512960t^{10} - 66244608t^{9} - 181878912t^{8} -
342337536t^{7} - 464624640t^{6} - 459399168t^{5} - 327932928t^{4} -
164339712t^{3} - 54521856t^{2} - 10616832t - 912384\]
\[B(t) = 81648t^{24} + 1057536t^{23} + 4510080t^{22} - 3760128t^{21} -
127516032t^{20} - 694987776t^{19} - 2332786176t^{18} - 5625262080t^{17} -
7988150016t^{16} + 11852144640t^{15} + 135410724864t^{14} + 552887451648t^{13} +
1524493135872t^{12} + 3159826661376t^{11} + 5125165498368t^{10} +
6628233314304t^{9} + 6886603100160t^{8} + 5744592617472t^{7} +
3816882929664t^{6} + 1988744970240t^{5} + 792581013504t^{4} + 232459075584t^{3}
+ 47106883584t^{2} + 5860491264t + 334430208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 8938028x + 9700468848$, with conductor $168640$ |
| Generic density of odd order reductions |
$13411/86016$ |