Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 2 + \left(30 a + 24\right)\cdot 47 + \left(7 a + 25\right)\cdot 47^{2} + \left(17 a + 14\right)\cdot 47^{3} + \left(43 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 28\cdot 47 + 45\cdot 47^{2} + 4\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 + 31\cdot 47 + 45\cdot 47^{2} + 10\cdot 47^{3} + 28\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 42 + \left(16 a + 17\right)\cdot 47 + \left(39 a + 10\right)\cdot 47^{2} + \left(29 a + 41\right)\cdot 47^{3} + \left(3 a + 27\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 24 + \left(45 a + 37\right)\cdot 47 + \left(23 a + 26\right)\cdot 47^{2} + \left(14 a + 39\right)\cdot 47^{3} + \left(17 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 46 + \left(a + 22\right)\cdot 47 + \left(23 a + 29\right)\cdot 47^{2} + \left(32 a + 44\right)\cdot 47^{3} + \left(29 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 8 a + 27 + \left(29 a + 11\right)\cdot 47 + \left(19 a + 44\right)\cdot 47^{2} + \left(14 a + 34\right)\cdot 47^{3} + \left(42 a + 4\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 39 a + 43 + \left(17 a + 14\right)\cdot 47 + \left(27 a + 7\right)\cdot 47^{2} + \left(32 a + 44\right)\cdot 47^{3} + \left(4 a + 27\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,7,6)(2,8,3,4)$ |
| $(1,7)(4,6)(5,8)$ |
| $(1,6,8)(4,7,5)$ |
| $(1,7)(2,3)(4,8)(5,6)$ |
| $(1,3,7,2)(4,5,8,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $-2$ |
| $12$ | $2$ | $(1,7)(4,6)(5,8)$ | $0$ |
| $8$ | $3$ | $(1,3,5)(2,6,7)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,2)(4,5,8,6)$ | $0$ |
| $8$ | $6$ | $(1,5,8,7,6,4)(2,3)$ | $1$ |
| $6$ | $8$ | $(1,6,2,8,7,5,3,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,5,2,4,7,6,3,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
