| Curve name |
$X_{192h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 9 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{192}$ |
| Curves that $X_{192h}$ minimally covers |
|
| Curves that minimally cover $X_{192h}$ |
|
| Curves that minimally cover $X_{192h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} - 23760t^{22} + 870048t^{20} + 2025216t^{18} -
276804864t^{16} + 1757306880t^{14} - 6309494784t^{12} + 28116910080t^{10} -
70862045184t^{8} + 8295284736t^{6} + 57019465728t^{4} - 24914165760t^{2} -
452984832\]
\[B(t) = 54t^{36} - 115344t^{34} - 1858464t^{32} + 510879744t^{30} -
15214086144t^{28} + 140525715456t^{26} - 307830620160t^{24} +
7470640005120t^{22} - 110151479328768t^{20} + 632942965555200t^{18} -
1762423669260288t^{16} + 1912483841310720t^{14} - 1260874220175360t^{12} +
9209493288124416t^{10} - 15953125592530944t^{8} + 8571139815112704t^{6} -
498877631299584t^{4} - 495398707789824t^{2} + 3710851743744\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 5882451x + 5488977549$, with conductor
$1470$ |
| Generic density of odd order reductions |
$11/112$ |