Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 69\cdot 71 + 34\cdot 71^{2} + 47\cdot 71^{3} + 31\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 19\cdot 71 + 5\cdot 71^{2} + 64\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 28\cdot 71 + 39\cdot 71^{2} + 21\cdot 71^{3} + 46\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 46\cdot 71 + 35\cdot 71^{2} + 65\cdot 71^{3} + 23\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 24\cdot 71 + 35\cdot 71^{2} + 5\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 42\cdot 71 + 31\cdot 71^{2} + 49\cdot 71^{3} + 24\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 + 51\cdot 71 + 65\cdot 71^{2} + 6\cdot 71^{3} + 48\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 62 + 71 + 36\cdot 71^{2} + 23\cdot 71^{3} + 39\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,3,7)(2,8,4,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,5,3,7)(2,8,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
