| Curve name |
$X_{82c}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 15 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 10 \\ 14 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{82}$ |
| Curves that $X_{82c}$ minimally covers |
|
| Curves that minimally cover $X_{82c}$ |
|
| Curves that minimally cover $X_{82c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1890t^{12} - 24624t^{11} - 137592t^{10} - 453600t^{9} - 1046520t^{8} -
1897344t^{7} - 2894400t^{6} - 3794688t^{5} - 4186080t^{4} - 3628800t^{3} -
2201472t^{2} - 787968t - 120960\]
\[B(t) = 31536t^{18} + 614304t^{17} + 5393952t^{16} + 28646784t^{15} +
104302080t^{14} + 278256384t^{13} + 558157824t^{12} + 820772352t^{11} +
755910144t^{10} - 1511820288t^{8} - 3283089408t^{7} - 4465262592t^{6} -
4452102144t^{5} - 3337666560t^{4} - 1833394176t^{3} - 690425856t^{2} -
157261824t - 16146432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 23x + 51$, with conductor $768$ |
| Generic density of odd order reductions |
$417/1792$ |