Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 a + 26 + \left(45 a + 27\right)\cdot 47 + \left(44 a + 17\right)\cdot 47^{2} + \left(28 a + 4\right)\cdot 47^{3} + \left(39 a + 17\right)\cdot 47^{4} + \left(36 a + 42\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 + 23\cdot 47 + 26\cdot 47^{2} + 17\cdot 47^{3} + 5\cdot 47^{4} + 26\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 2 + \left(35 a + 15\right)\cdot 47 + \left(19 a + 2\right)\cdot 47^{2} + \left(29 a + 21\right)\cdot 47^{3} + \left(46 a + 24\right)\cdot 47^{4} + \left(26 a + 26\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 43\cdot 47 + 37\cdot 47^{2} + 4\cdot 47^{3} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 30 + \left(11 a + 24\right)\cdot 47 + \left(27 a + 6\right)\cdot 47^{2} + \left(17 a + 13\right)\cdot 47^{3} + 41\cdot 47^{4} + \left(20 a + 33\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 a + 31 + \left(a + 22\right)\cdot 47 + 18\cdot 47^{2} + \left(22 a + 9\right)\cdot 47^{3} + \left(20 a + 30\right)\cdot 47^{4} + \left(3 a + 21\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 3 a + 25 + \left(45 a + 29\right)\cdot 47 + \left(46 a + 16\right)\cdot 47^{2} + \left(24 a + 6\right)\cdot 47^{3} + \left(26 a + 2\right)\cdot 47^{4} + \left(43 a + 8\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 24 a + 25 + \left(a + 1\right)\cdot 47 + \left(2 a + 15\right)\cdot 47^{2} + \left(18 a + 17\right)\cdot 47^{3} + \left(7 a + 20\right)\cdot 47^{4} + \left(10 a + 29\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,8,6)(2,7,4,3)$ |
| $(1,5)(2,4)(6,8)$ |
| $(1,6,2)(4,8,5)$ |
| $(1,3,8,7)(2,5,4,6)$ |
| $(1,8)(2,4)(3,7)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,4)(6,8)$ | $0$ |
| $8$ | $3$ | $(2,5,3)(4,6,7)$ | $-1$ |
| $6$ | $4$ | $(1,5,8,6)(2,7,4,3)$ | $0$ |
| $8$ | $6$ | $(1,8)(2,7,5,4,3,6)$ | $1$ |
| $6$ | $8$ | $(1,4,5,3,8,2,6,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,2,5,7,8,4,6,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
