Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{3} + 3 x + 42 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 a^{2} + 29 a + 26 + \left(4 a^{2} + 2 a + 35\right)\cdot 47 + \left(23 a^{2} + 30 a + 11\right)\cdot 47^{2} + \left(30 a^{2} + 37 a + 35\right)\cdot 47^{3} + \left(3 a^{2} + 20 a + 7\right)\cdot 47^{4} + \left(45 a^{2} + 30 a + 8\right)\cdot 47^{5} + \left(24 a^{2} + 33 a + 20\right)\cdot 47^{6} + \left(23 a^{2} + 9 a + 4\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 + 6\cdot 47 + 34\cdot 47^{2} + 15\cdot 47^{3} + 16\cdot 47^{4} + 25\cdot 47^{5} + 43\cdot 47^{6} + 12\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a^{2} + 37 a + 28 + \left(26 a^{2} + 4 a + 21\right)\cdot 47 + \left(3 a^{2} + 40 a + 33\right)\cdot 47^{2} + \left(27 a^{2} + 40 a + 12\right)\cdot 47^{3} + \left(41 a^{2} + 33 a + 32\right)\cdot 47^{4} + \left(37 a^{2} + 42 a + 6\right)\cdot 47^{5} + \left(3 a^{2} + 34 a + 25\right)\cdot 47^{6} + \left(6 a^{2} + 43 a + 7\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 a^{2} + 12 a + 9 + \left(9 a^{2} + 19 a + 46\right)\cdot 47 + \left(42 a^{2} + 3 a + 2\right)\cdot 47^{2} + \left(23 a^{2} + 33 a + 22\right)\cdot 47^{3} + \left(18 a^{2} + 10 a + 37\right)\cdot 47^{4} + \left(18 a^{2} + 41 a + 1\right)\cdot 47^{5} + \left(35 a^{2} + 33 a + 41\right)\cdot 47^{6} + \left(12 a^{2} + 25 a + 29\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 6\cdot 47 + 37\cdot 47^{2} + 44\cdot 47^{3} + 40\cdot 47^{4} + 4\cdot 47^{5} + 40\cdot 47^{6} + 34\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 a^{2} + 6 a + 16 + \left(32 a^{2} + 25 a + 44\right)\cdot 47 + \left(28 a^{2} + 13 a + 22\right)\cdot 47^{2} + \left(39 a^{2} + 23 a + 6\right)\cdot 47^{3} + \left(24 a^{2} + 15 a + 3\right)\cdot 47^{4} + \left(30 a^{2} + 22 a + 26\right)\cdot 47^{5} + \left(33 a^{2} + 26 a + 37\right)\cdot 47^{6} + \left(10 a^{2} + 11 a + 25\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 15 a^{2} + 27 a + 38 + \left(42 a^{2} + 5 a + 6\right)\cdot 47 + \left(35 a^{2} + 16 a + 4\right)\cdot 47^{2} + \left(45 a^{2} + 46 a + 3\right)\cdot 47^{3} + \left(30 a^{2} + 46 a + 11\right)\cdot 47^{4} + \left(7 a^{2} + 13 a + 40\right)\cdot 47^{5} + \left(8 a^{2} + 19 a + 33\right)\cdot 47^{6} + \left(30 a^{2} + 28 a + 8\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 22 a^{2} + 30 a + 5 + \left(25 a^{2} + 36 a + 20\right)\cdot 47 + \left(7 a^{2} + 37 a + 41\right)\cdot 47^{2} + \left(21 a^{2} + 6 a\right)\cdot 47^{3} + \left(21 a^{2} + 13 a + 39\right)\cdot 47^{4} + \left(a^{2} + 37 a + 27\right)\cdot 47^{5} + \left(35 a^{2} + 39 a + 40\right)\cdot 47^{6} + \left(10 a^{2} + 21 a + 16\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,7,6)(4,8,5)$ |
| $(1,6,3,8)(2,4,5,7)$ |
| $(1,4,3,7)(2,8,5,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
$-2$ |
| $4$ |
$3$ |
$(2,7,6)(4,8,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $4$ |
$3$ |
$(2,6,7)(4,5,8)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $6$ |
$4$ |
$(1,4,3,7)(2,8,5,6)$ |
$0$ |
$0$ |
| $4$ |
$6$ |
$(1,4,5,3,7,2)(6,8)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $4$ |
$6$ |
$(1,2,7,3,5,4)(6,8)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
