| Curve name |
$X_{226c}$ |
| 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} 7 & 7 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{226}$ |
| Curves that $X_{226c}$ minimally covers |
|
| Curves that minimally cover $X_{226c}$ |
|
| Curves that minimally cover $X_{226c}$ 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 |
$y^2 = x^3 - 2234507x - 1212558606$, with conductor $84320$ |
| Generic density of odd order reductions |
$9827/86016$ |