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:
| $r_{ 1 }$ |
$=$ |
$ 37 a + 31 + \left(34 a + 5\right)\cdot 47 + \left(4 a + 25\right)\cdot 47^{2} + \left(27 a + 24\right)\cdot 47^{3} + \left(33 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 20 + \left(46 a + 32\right)\cdot 47 + \left(39 a + 37\right)\cdot 47^{2} + \left(6 a + 34\right)\cdot 47^{3} + \left(7 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 11 + \left(12 a + 38\right)\cdot 47 + \left(42 a + 46\right)\cdot 47^{2} + \left(19 a + 26\right)\cdot 47^{3} + \left(13 a + 35\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 30 + 25\cdot 47 + \left(7 a + 24\right)\cdot 47^{2} + \left(40 a + 8\right)\cdot 47^{3} + \left(39 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 44 + \left(19 a + 21\right)\cdot 47 + \left(21 a + 17\right)\cdot 47^{2} + \left(15 a + 45\right)\cdot 47^{3} + \left(3 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 19\cdot 47 + 34\cdot 47^{2} + 23\cdot 47^{3} + 39\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 a + 13 + \left(27 a + 5\right)\cdot 47 + \left(25 a + 41\right)\cdot 47^{2} + \left(31 a + 7\right)\cdot 47^{3} + \left(43 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 6 + 39\cdot 47 + 7\cdot 47^{2} + 16\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,5,6)(2,3,4,7)$ |
| $(1,5)(2,4)(3,7)(6,8)$ |
| $(1,3,6)(5,7,8)$ |
| $(1,5)(3,8)(6,7)$ |
| $(1,4,5,2)(3,6,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,7)(6,8)$ |
$-4$ |
| $12$ |
$2$ |
$(1,5)(3,8)(6,7)$ |
$0$ |
| $8$ |
$3$ |
$(1,4,8)(2,6,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,4,5,2)(3,6,7,8)$ |
$0$ |
| $8$ |
$6$ |
$(1,6,4,5,8,2)(3,7)$ |
$-1$ |
| $6$ |
$8$ |
$(1,6,2,3,5,8,4,7)$ |
$0$ |
| $6$ |
$8$ |
$(1,8,2,7,5,6,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
