| Curve name |
$X_{38a}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 2 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{38}$ |
| Curves that $X_{38a}$ minimally covers |
|
| Curves that minimally cover $X_{38a}$ |
|
| Curves that minimally cover $X_{38a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{8} + 162t^{6} - 432t^{4} + 648t^{2} - 432\]
\[B(t) = 54t^{12} - 486t^{10} + 1296t^{8} - 5184t^{4} + 7776t^{2} - 3456\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 931x - 10290$, with conductor $392$ |
| Generic density of odd order reductions |
$643/5376$ |