Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 60\cdot 71 + 4\cdot 71^{2} + 25\cdot 71^{3} + 11\cdot 71^{4} + 30\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 62\cdot 71 + 51\cdot 71^{2} + 50\cdot 71^{3} + 61\cdot 71^{4} + 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 7\cdot 71 + 33\cdot 71^{2} + 25\cdot 71^{3} + 19\cdot 71^{4} + 55\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 63\cdot 71 + 68\cdot 71^{2} + 17\cdot 71^{3} + 29\cdot 71^{4} + 35\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 18\cdot 71 + 54\cdot 71^{2} + 6\cdot 71^{3} + 51\cdot 71^{4} + 17\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 23\cdot 71 + 71^{2} + 27\cdot 71^{3} + 46\cdot 71^{4} + 18\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 70\cdot 71 + 13\cdot 71^{2} + 21\cdot 71^{3} + 50\cdot 71^{4} + 58\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 48\cdot 71 + 55\cdot 71^{2} + 38\cdot 71^{3} + 14\cdot 71^{4} + 66\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,6)(3,8)(4,5)$ |
| $(1,4)(3,8)(5,7)$ |
| $(1,5,7,4)(2,3,6,8)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,6)(3,8)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,8)(6,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(3,8)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,5,7,4)(2,3,6,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,4,8,7,6,5,3)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,5,2,7,3,4,6)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
