Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 9\cdot 71 + 35\cdot 71^{2} + 25\cdot 71^{3} + 19\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 8\cdot 71 + 8\cdot 71^{2} + 38\cdot 71^{3} + 68\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 19\cdot 71 + 20\cdot 71^{2} + 20\cdot 71^{3} + 3\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 55\cdot 71 + 60\cdot 71^{2} + 21\cdot 71^{3} + 48\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 35\cdot 71 + 33\cdot 71^{2} + 32\cdot 71^{3} + 15\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 16\cdot 71 + 70\cdot 71^{2} + 54\cdot 71^{3} + 14\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 52 + 40\cdot 71 + 41\cdot 71^{2} + 35\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 + 27\cdot 71 + 14\cdot 71^{2} + 55\cdot 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 |
| $(1,4)(2,8)(3,7)(5,6)$ |
| $(1,5)(2,8)(4,6)$ |
| $(1,6,4,5)(2,3,8,7)$ |
| $(1,3,4,7)(2,5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,8)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,6,4,5)(2,3,8,7)$ | $0$ |
| $4$ | $4$ | $(1,3,4,7)(2,5,8,6)$ | $0$ |
| $2$ | $8$ | $(1,3,6,8,4,7,5,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,7,6,2,4,3,5,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
