Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 40\cdot 71 + 31\cdot 71^{2} + 11\cdot 71^{3} + 67\cdot 71^{4} + 50\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 47\cdot 71 + 21\cdot 71^{2} + 7\cdot 71^{3} + 63\cdot 71^{4} + 21\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 29\cdot 71 + 61\cdot 71^{2} + 3\cdot 71^{3} + 44\cdot 71^{4} + 19\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 5\cdot 71 + 26\cdot 71^{2} + 52\cdot 71^{3} + 30\cdot 71^{4} + 14\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 23\cdot 71 + 58\cdot 71^{2} + 2\cdot 71^{3} + 44\cdot 71^{4} + 61\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 66\cdot 71 + 22\cdot 71^{2} + 3\cdot 71^{3} + 57\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 48\cdot 71 + 62\cdot 71^{2} + 70\cdot 71^{3} + 51\cdot 71^{4} + 54\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 70 + 21\cdot 71 + 70\cdot 71^{2} + 60\cdot 71^{3} + 53\cdot 71^{4} + 3\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,4)(3,5)(6,8)$ |
| $(1,8)(3,6)(4,5)$ |
| $(1,5,4,8)(2,3,7,6)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,7)(2,4)(3,5)(6,8)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(3,6)(4,5)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,5,4,8)(2,3,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,8,3,4,2,5,6)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,5,7,4,6,8,2)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
