| Curve name |
$X_{75i}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{75i}$ minimally covers |
|
| Curves that minimally cover $X_{75i}$ |
|
| Curves that minimally cover $X_{75i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{8} + 829440t^{6} - 231552t^{4} + 12960t^{2} - 27\]
\[B(t) = 14155776t^{12} + 222953472t^{10} - 230252544t^{8} + 48771072t^{6} -
3597696t^{4} + 54432t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 13936x - 212640$, with conductor $1848$ |
| Generic density of odd order reductions |
$65/896$ |