| Curve name |
$X_{230f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{230}$ |
| Curves that $X_{230f}$ minimally covers |
|
| Curves that minimally cover $X_{230f}$ |
|
| Curves that minimally cover $X_{230f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27648t^{16} + 110592t^{14} - 82944t^{12} - 27648t^{10} + 17280t^{8} -
6912t^{6} - 5184t^{4} + 1728t^{2} - 108\]
\[B(t) = -1769472t^{24} + 10616832t^{22} - 18579456t^{20} + 6193152t^{18} +
6303744t^{16} - 3981312t^{14} - 774144t^{12} - 995328t^{10} + 393984t^{8} +
96768t^{6} - 72576t^{4} + 10368t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 257x - 3423$, with conductor $1344$ |
| Generic density of odd order reductions |
$271/2688$ |