| Curve name |
$X_{226d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 11 & 11 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 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 $X_{226d}$ minimally covers |
|
| Curves that minimally cover $X_{226d}$ |
|
| Curves that minimally cover $X_{226d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -891t^{16} - 7776t^{15} - 18576t^{14} + 38016t^{13} + 267408t^{12} +
107136t^{11} - 3378240t^{10} - 16561152t^{9} - 45469728t^{8} - 85584384t^{7} -
116156160t^{6} - 114849792t^{5} - 81983232t^{4} - 41084928t^{3} - 13630464t^{2}
- 2654208t - 228096\]
\[B(t) = 10206t^{24} + 132192t^{23} + 563760t^{22} - 470016t^{21} -
15939504t^{20} - 86873472t^{19} - 291598272t^{18} - 703157760t^{17} -
998518752t^{16} + 1481518080t^{15} + 16926340608t^{14} + 69110931456t^{13} +
190561641984t^{12} + 394978332672t^{11} + 640645687296t^{10} + 828529164288t^{9}
+ 860825387520t^{8} + 718074077184t^{7} + 477110366208t^{6} + 248593121280t^{5}
+ 99072626688t^{4} + 29057384448t^{3} + 5888360448t^{2} + 732561408t +
41803776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 6434507x + 6281938894$, with conductor $84320$ |
| Generic density of odd order reductions |
$9827/86016$ |