| Curve name |
$X_{20a}$ |
| Index |
$16$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 3 & 0 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 3 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{20}$ |
| Curves that $X_{20a}$ minimally covers |
|
| Curves that minimally cover $X_{20a}$ |
|
| Curves that minimally cover $X_{20a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{8} + 2592t^{7} - 16848t^{6} - 9504t^{5} + 240408t^{4} +
318816t^{3} - 422928t^{2} - 940896t - 431244\]
\[B(t) = 432t^{12} - 15552t^{11} + 194400t^{10} - 782784t^{9} - 2010096t^{8} +
13779072t^{7} + 38574144t^{6} - 69579648t^{5} - 514588464t^{4} - 1123581888t^{3}
- 1256096160t^{2} - 724489920t - 170772624\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 12x + 112$, with conductor $5184$ |
| Generic density of odd order reductions |
$395/896$ |