| Curve name |
$X_{75d}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{75d}$ minimally covers |
|
| Curves that minimally cover $X_{75d}$ |
|
| Curves that minimally cover $X_{75d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{16} + 3397386240t^{14} - 934281216t^{12} - 53084160t^{10}
+ 29417472t^{8} - 829440t^{6} - 228096t^{4} + 12960t^{2} - 27\]
\[B(t) = 3710851743744t^{24} + 58445914963968t^{22} - 60533269069824t^{20} +
10045391634432t^{18} + 1888946749440t^{16} - 542222843904t^{14} +
8472231936t^{10} - 461168640t^{8} - 38320128t^{6} + 3608064t^{4} - 54432t^{2} -
54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 82624568x - 99225432788$, with conductor $142296$ |
| Generic density of odd order reductions |
$307/2688$ |