Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 9\cdot 71 + 17\cdot 71^{2} + 20\cdot 71^{3} + 58\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 14\cdot 71 + 5\cdot 71^{2} + 49\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 40\cdot 71 + 35\cdot 71^{2} + 13\cdot 71^{3} + 16\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 25\cdot 71 + 47\cdot 71^{2} + 28\cdot 71^{3} + 65\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 45\cdot 71 + 23\cdot 71^{2} + 42\cdot 71^{3} + 5\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 30\cdot 71 + 35\cdot 71^{2} + 57\cdot 71^{3} + 54\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 56\cdot 71 + 65\cdot 71^{2} + 21\cdot 71^{3} + 23\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 61\cdot 71 + 53\cdot 71^{2} + 50\cdot 71^{3} + 12\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.
