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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 12 + \left(45 a + 27\right)\cdot 47 + \left(22 a + 1\right)\cdot 47^{2} + \left(39 a + 3\right)\cdot 47^{3} + 4 a\cdot 47^{4} + 45\cdot 47^{5} + \left(12 a + 29\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 18 + \left(14 a + 14\right)\cdot 47 + \left(10 a + 37\right)\cdot 47^{2} + \left(18 a + 32\right)\cdot 47^{3} + \left(15 a + 23\right)\cdot 47^{4} + \left(23 a + 26\right)\cdot 47^{5} + \left(35 a + 25\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a + 25 + \left(32 a + 16\right)\cdot 47 + \left(36 a + 43\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(31 a + 36\right)\cdot 47^{4} + \left(23 a + 10\right)\cdot 47^{5} + \left(11 a + 26\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 27 + \left(a + 40\right)\cdot 47 + \left(24 a + 1\right)\cdot 47^{2} + \left(7 a + 12\right)\cdot 47^{3} + \left(42 a + 17\right)\cdot 47^{4} + \left(46 a + 40\right)\cdot 47^{5} + \left(34 a + 6\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a + 10 + \left(15 a + 3\right)\cdot 47 + \left(34 a + 2\right)\cdot 47^{2} + \left(25 a + 32\right)\cdot 47^{3} + \left(10 a + 10\right)\cdot 47^{4} + \left(23 a + 38\right)\cdot 47^{5} + \left(23 a + 37\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 2 + \left(31 a + 39\right)\cdot 47 + \left(12 a + 7\right)\cdot 47^{2} + \left(21 a + 2\right)\cdot 47^{3} + \left(36 a + 6\right)\cdot 47^{4} + \left(23 a + 27\right)\cdot 47^{5} + \left(23 a + 14\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)(2,6,4)$ |
| $(1,5,3)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
$0$ |
| $1$ |
$3$ |
$(1,3,5)(2,6,4)$ |
$2 \zeta_{3}$ |
$-2 \zeta_{3} - 2$ |
| $1$ |
$3$ |
$(1,5,3)(2,4,6)$ |
$-2 \zeta_{3} - 2$ |
$2 \zeta_{3}$ |
| $2$ |
$3$ |
$(1,5,3)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $2$ |
$3$ |
$(1,3,5)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $2$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$-1$ |
$-1$ |
| $3$ |
$6$ |
$(1,6,5,2,3,4)$ |
$0$ |
$0$ |
| $3$ |
$6$ |
$(1,4,3,2,5,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
