| Curve name |
$X_{227f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227f}$ minimally covers |
|
| Curves that minimally cover $X_{227f}$ |
|
| Curves that minimally cover $X_{227f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{24} + 13589544960t^{22} - 15344861184t^{20} +
7644119040t^{18} - 3848601600t^{16} + 1167851520t^{14} - 361414656t^{12} +
72990720t^{10} - 15033600t^{8} + 1866240t^{6} - 234144t^{4} + 12960t^{2} - 27\]
\[B(t) = 3710851743744t^{36} + 233783659855872t^{34} - 963661812203520t^{32} +
935134639423488t^{30} - 663430713311232t^{28} + 347022620098560t^{26} -
154536681406464t^{24} + 54793045278720t^{22} - 17665389232128t^{20} +
4587490639872t^{18} - 1104086827008t^{16} + 214035333120t^{14} -
37728681984t^{12} + 5295144960t^{10} - 632696832t^{8} + 55738368t^{6} -
3589920t^{4} + 54432t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + 1892944x - 4740612030$, with conductor $8670$ |
| Generic density of odd order reductions |
$271/2688$ |