Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 6\cdot 71 + 37\cdot 71^{2} + 28\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 47\cdot 71 + 32\cdot 71^{2} + 46\cdot 71^{3} + 8\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 13\cdot 71 + 45\cdot 71^{2} + 64\cdot 71^{3} + 25\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 + 37\cdot 71 + 3\cdot 71^{2} + 45\cdot 71^{3} + 41\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 33\cdot 71 + 67\cdot 71^{2} + 25\cdot 71^{3} + 29\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 57\cdot 71 + 25\cdot 71^{2} + 6\cdot 71^{3} + 45\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 23\cdot 71 + 38\cdot 71^{2} + 24\cdot 71^{3} + 62\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 64\cdot 71 + 33\cdot 71^{2} + 42\cdot 71^{3} + 48\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,4)(2,5,8,6)$ |
| $(1,2)(3,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,4)(2,5,8,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
