| Curve name |
$X_{188f}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{188}$ |
| Curves that $X_{188f}$ minimally covers |
|
| Curves that minimally cover $X_{188f}$ |
|
| Curves that minimally cover $X_{188f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} + 4320t^{22} - 63072t^{20} + 390528t^{18} - 1187136t^{16} +
8985600t^{14} - 59249664t^{12} + 35942400t^{10} - 18994176t^{8} + 24993792t^{6}
- 16146432t^{4} + 4423680t^{2} - 442368\]
\[B(t) = -432t^{36} + 25920t^{34} - 637632t^{32} + 8183808t^{30} -
53498880t^{28} + 39481344t^{26} + 2402168832t^{24} - 19503120384t^{22} +
56518041600t^{20} - 49236443136t^{18} + 226072166400t^{16} - 312049926144t^{14}
+ 153738805248t^{12} + 10107224064t^{10} - 54782853120t^{8} + 33520877568t^{6} -
10446962688t^{4} + 1698693120t^{2} - 113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 263489x + 51927231$, with conductor $9408$ |
| Generic density of odd order reductions |
$271/2688$ |