| Curve name |
$X_{13g}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{13}$ |
| Curves that $X_{13g}$ minimally covers |
|
| Curves that minimally cover $X_{13g}$ |
|
| Curves that minimally cover $X_{13g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{4} + 1728t^{3} - 1728t^{2} - 82944t + 331776\]
\[B(t) = -432t^{6} + 10368t^{5} - 51840t^{4} - 525312t^{3} + 5971968t^{2} -
15925248t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 69x - 194$, with conductor $30$ |
| Generic density of odd order reductions |
$513/3584$ |