Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 24\cdot 71 + 60\cdot 71^{2} + 19\cdot 71^{3} + 63\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 32\cdot 71 + 28\cdot 71^{2} + 44\cdot 71^{3} + 41\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 16\cdot 71 + 70\cdot 71^{2} + 29\cdot 71^{3} + 66\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 12\cdot 71 + 69\cdot 71^{2} + 47\cdot 71^{3} + 38\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 + 69\cdot 71 + 53\cdot 71^{2} + 47\cdot 71^{3} + 41\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 + 43\cdot 71 + 19\cdot 71^{2} + 16\cdot 71^{3} + 66\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 + 57\cdot 71 + 62\cdot 71^{2} + 4\cdot 71^{3} + 17\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 27\cdot 71 + 61\cdot 71^{2} + 71^{3} + 20\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,6)(4,8)(5,7)$ |
| $(1,2,8,7)(3,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
