| Curve name |
$X_{230e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{230}$ |
| Curves that $X_{230e}$ minimally covers |
|
| Curves that minimally cover $X_{230e}$ |
|
| Curves that minimally cover $X_{230e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -6912t^{16} + 27648t^{14} - 20736t^{12} - 6912t^{10} + 4320t^{8} -
1728t^{6} - 1296t^{4} + 432t^{2} - 27\]
\[B(t) = 221184t^{24} - 1327104t^{22} + 2322432t^{20} - 774144t^{18} -
787968t^{16} + 497664t^{14} + 96768t^{12} + 124416t^{10} - 49248t^{8} -
12096t^{6} + 9072t^{4} - 1296t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 4x + 5$, with conductor $42$ |
| Generic density of odd order reductions |
$53/896$ |