| Curve name |
$X_{58d}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 4 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{58}$ |
| Curves that $X_{58d}$ minimally covers |
|
| Curves that minimally cover $X_{58d}$ |
|
| Curves that minimally cover $X_{58d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{14} + 216t^{12} - 1620t^{10} + 3024t^{8} - 1620t^{6} + 216t^{4}
- 108t^{2}\]
\[B(t) = 432t^{21} - 1296t^{19} - 12960t^{17} + 42336t^{15} - 57024t^{13} +
57024t^{11} - 42336t^{9} + 12960t^{7} + 1296t^{5} - 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 117005x + 14142372$, with conductor $2925$ |
| Generic density of odd order reductions |
$307/2688$ |