Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 7\cdot 71 + 39\cdot 71^{3} + 29\cdot 71^{4} + 7\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 12\cdot 71 + 19\cdot 71^{2} + 32\cdot 71^{3} + 3\cdot 71^{4} + 64\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 68\cdot 71 + 35\cdot 71^{2} + 4\cdot 71^{3} + 16\cdot 71^{4} + 6\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 7\cdot 71 + 9\cdot 71^{2} + 27\cdot 71^{3} + 64\cdot 71^{4} + 46\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 63\cdot 71 + 61\cdot 71^{2} + 43\cdot 71^{3} + 6\cdot 71^{4} + 24\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 2\cdot 71 + 35\cdot 71^{2} + 66\cdot 71^{3} + 54\cdot 71^{4} + 64\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 58\cdot 71 + 51\cdot 71^{2} + 38\cdot 71^{3} + 67\cdot 71^{4} + 6\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 68 + 63\cdot 71 + 70\cdot 71^{2} + 31\cdot 71^{3} + 41\cdot 71^{4} + 63\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
