Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 14 + \left(8 a + 1\right)\cdot 47 + \left(22 a + 17\right)\cdot 47^{2} + \left(33 a + 10\right)\cdot 47^{3} + \left(4 a + 14\right)\cdot 47^{4} + \left(10 a + 7\right)\cdot 47^{5} + \left(34 a + 25\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 45 + \left(38 a + 26\right)\cdot 47 + \left(24 a + 5\right)\cdot 47^{2} + \left(13 a + 8\right)\cdot 47^{3} + \left(42 a + 37\right)\cdot 47^{4} + \left(36 a + 22\right)\cdot 47^{5} + \left(12 a + 36\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 9 + \left(46 a + 34\right)\cdot 47 + \left(37 a + 3\right)\cdot 47^{2} + \left(23 a + 36\right)\cdot 47^{3} + \left(19 a + 34\right)\cdot 47^{4} + \left(a + 9\right)\cdot 47^{5} + \left(12 a + 40\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 21 + \left(8 a + 19\right)\cdot 47 + \left(13 a + 8\right)\cdot 47^{2} + \left(10 a + 40\right)\cdot 47^{3} + \left(24 a + 6\right)\cdot 47^{4} + \left(11 a + 31\right)\cdot 47^{5} + \left(46 a + 41\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 24 + \left(38 a + 11\right)\cdot 47 + \left(33 a + 26\right)\cdot 47^{2} + 36 a\cdot 47^{3} + \left(22 a + 45\right)\cdot 47^{4} + \left(35 a + 29\right)\cdot 47^{5} + 28\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 28 + \left(9 a + 33\right)\cdot 47^{2} + \left(23 a + 45\right)\cdot 47^{3} + \left(27 a + 2\right)\cdot 47^{4} + \left(45 a + 40\right)\cdot 47^{5} + \left(34 a + 15\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,3)$ |
| $(1,4)(2,3)(5,6)$ |
| $(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,5,3)(2,4,6)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,3,5)(2,6,4)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(1,5,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,3,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $2$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$-1$ |
$-1$ |
| $3$ |
$6$ |
$(1,6,5,2,3,4)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,4,3,2,5,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
