Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 36\cdot 71 + 43\cdot 71^{2} + 55\cdot 71^{3} + 68\cdot 71^{4} + 28\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 58\cdot 71 + 59\cdot 71^{2} + 14\cdot 71^{3} + 26\cdot 71^{4} + 9\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 34\cdot 71 + 71^{2} + 9\cdot 71^{3} + 30\cdot 71^{4} + 49\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 43\cdot 71 + 46\cdot 71^{2} + 35\cdot 71^{3} + 11\cdot 71^{4} + 40\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 45\cdot 71 + 22\cdot 71^{2} + 61\cdot 71^{3} + 61\cdot 71^{4} + 11\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 34\cdot 71 + 26\cdot 71^{2} + 17\cdot 71^{3} + 30\cdot 71^{4} + 66\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 12\cdot 71 + 68\cdot 71^{2} + 34\cdot 71^{3} + 65\cdot 71^{4} + 2\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 17\cdot 71 + 15\cdot 71^{2} + 55\cdot 71^{3} + 60\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 |
| $(1,4,6,2,7,3,8,5)$ |
| $(1,7)(2,5)(3,4)(6,8)$ |
| $(1,6,7,8)(2,3,5,4)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,5)(3,4)(6,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(3,4)(6,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,6,7,8)(2,3,5,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,8,3,7,2,6,4)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,6,5,7,4,8,2)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
