Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 65\cdot 71 + 51\cdot 71^{2} + 46\cdot 71^{3} + 14\cdot 71^{4} + 61\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 63\cdot 71 + 51\cdot 71^{2} + 35\cdot 71^{3} + 32\cdot 71^{4} + 56\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 30\cdot 71 + 25\cdot 71^{2} + 33\cdot 71^{3} + 66\cdot 71^{4} + 9\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 35\cdot 71 + 56\cdot 71^{2} + 54\cdot 71^{3} + 22\cdot 71^{4} + 20\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 + 62\cdot 71 + 40\cdot 71^{3} + 59\cdot 71^{4} + 40\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 55\cdot 71 + 71^{2} + 36\cdot 71^{3} + 15\cdot 71^{4} + 19\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 13\cdot 71 + 59\cdot 71^{2} + 13\cdot 71^{3} + 64\cdot 71^{4} + 70\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 + 28\cdot 71 + 36\cdot 71^{2} + 23\cdot 71^{3} + 8\cdot 71^{4} + 5\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(5,7)(6,8)$ |
| $(3,4)(6,8)$ |
| $(1,6,2,8)(3,7,4,5)$ |
| $(1,5,6,3,2,7,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-4$ |
| $2$ | $2$ | $(3,4)(5,7)$ | $0$ |
| $4$ | $2$ | $(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,8)(3,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,2,8)(3,7,4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,2,8)(3,5,4,7)$ | $0$ |
| $4$ | $8$ | $(1,5,6,3,2,7,8,4)$ | $0$ |
| $4$ | $8$ | $(1,3,8,5,2,4,6,7)$ | $0$ |
| $4$ | $8$ | $(1,7,6,3,2,5,8,4)$ | $0$ |
| $4$ | $8$ | $(1,3,8,7,2,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
