| Curve name |
$X_{75c}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{75c}$ minimally covers |
|
| Curves that minimally cover $X_{75c}$ |
|
| Curves that minimally cover $X_{75c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7077888t^{12} + 51314688t^{10} - 1658880t^{8} - 2045952t^{6} -
25920t^{4} + 12528t^{2} - 27\]
\[B(t) = 7247757312t^{18} + 116870086656t^{16} - 74742497280t^{14} -
13872660480t^{12} + 2218917888t^{10} + 277364736t^{8} - 27095040t^{6} -
2280960t^{4} + 55728t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 3484077x + 2577412978$, with conductor $16560$ |
| Generic density of odd order reductions |
$635/5376$ |