| Curve name |
$X_{185c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{185}$ |
| Curves that $X_{185c}$ minimally covers |
|
| Curves that minimally cover $X_{185c}$ |
|
| Curves that minimally cover $X_{185c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7077888t^{32} - 106168320t^{28} - 58392576t^{24} + 6635520t^{20} +
7354368t^{16} + 414720t^{12} - 228096t^{8} - 25920t^{4} - 108\]
\[B(t) = 7247757312t^{48} - 228304355328t^{44} - 472916164608t^{40} -
156959244288t^{36} + 59029585920t^{32} + 33888927744t^{28} - 2118057984t^{20} -
230584320t^{16} + 38320128t^{12} + 7216128t^{8} + 217728t^{4} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1944300x + 1042958000$, with conductor $14400$ |
| Generic density of odd order reductions |
$51/448$ |