Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 8 + \left(17 a + 46\right)\cdot 47 + \left(43 a + 17\right)\cdot 47^{2} + \left(40 a + 30\right)\cdot 47^{3} + \left(44 a + 13\right)\cdot 47^{4} + \left(22 a + 24\right)\cdot 47^{5} + \left(34 a + 16\right)\cdot 47^{6} + \left(22 a + 14\right)\cdot 47^{7} + \left(26 a + 16\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 a + 17 + \left(a + 27\right)\cdot 47 + \left(38 a + 30\right)\cdot 47^{2} + \left(8 a + 26\right)\cdot 47^{3} + \left(26 a + 20\right)\cdot 47^{4} + \left(28 a + 3\right)\cdot 47^{5} + 10 a\cdot 47^{6} + \left(37 a + 28\right)\cdot 47^{7} + \left(34 a + 5\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 5\cdot 47 + 36\cdot 47^{2} + 41\cdot 47^{3} + 17\cdot 47^{4} + 44\cdot 47^{5} + 14\cdot 47^{6} + 7\cdot 47^{7} + 31\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 41 + \left(29 a + 41\right)\cdot 47 + \left(3 a + 39\right)\cdot 47^{2} + \left(6 a + 21\right)\cdot 47^{3} + \left(2 a + 15\right)\cdot 47^{4} + \left(24 a + 25\right)\cdot 47^{5} + \left(12 a + 15\right)\cdot 47^{6} + \left(24 a + 25\right)\cdot 47^{7} + \left(20 a + 46\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 46 + \left(45 a + 39\right)\cdot 47 + \left(8 a + 10\right)\cdot 47^{2} + \left(38 a + 6\right)\cdot 47^{3} + \left(20 a + 17\right)\cdot 47^{4} + \left(18 a + 34\right)\cdot 47^{5} + \left(36 a + 39\right)\cdot 47^{6} + \left(9 a + 44\right)\cdot 47^{7} + \left(12 a + 37\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 26\cdot 47 + 5\cdot 47^{2} + 14\cdot 47^{3} + 9\cdot 47^{4} + 9\cdot 47^{5} + 7\cdot 47^{6} + 21\cdot 47^{7} + 3\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(3,4)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,3)(2,6,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,6,4,5,3,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
