| Curve name |
$X_{233g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{233}$ |
| Curves that $X_{233g}$ minimally covers |
|
| Curves that minimally cover $X_{233g}$ |
|
| Curves that minimally cover $X_{233g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7421703487488t^{24} - 241205363343360t^{22} - 768378239188992t^{20} -
998045172891648t^{18} - 650064036888576t^{16} - 221609239511040t^{14} -
38741867495424t^{12} - 3462644367360t^{10} - 158707040256t^{8} - 3807240192t^{6}
- 45798912t^{4} - 224640t^{2} - 108\]
\[B(t) = -7782220156096217088t^{36} + 461096544248700862464t^{34} +
3693514644397228032000t^{32} + 10826770597788984016896t^{30} +
16776422303787594547200t^{28} + 15605925853478655098880t^{26} +
9184150139321494536192t^{24} + 3479631896514473754624t^{22} +
848542337948033482752t^{20} + 132582210750105255936t^{18} +
13258474030438023168t^{16} + 849519505984978944t^{14} + 35034752423559168t^{12}
+ 930185666887680t^{10} + 15624260812800t^{8} + 157550247936t^{6} +
839808000t^{4} + 1638144t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 513153409x + 4474407604513$, with conductor $55488$ |
| Generic density of odd order reductions |
$109/896$ |