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^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 42 + \left(4 a + 41\right)\cdot 47 + \left(24 a + 39\right)\cdot 47^{2} + \left(35 a + 30\right)\cdot 47^{3} + \left(7 a + 36\right)\cdot 47^{4} + \left(40 a + 41\right)\cdot 47^{5} + \left(20 a + 24\right)\cdot 47^{6} + \left(4 a + 41\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 + 8\cdot 47 + 10\cdot 47^{2} + 10\cdot 47^{3} + 11\cdot 47^{4} + 2\cdot 47^{5} + 30\cdot 47^{6} + 31\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 36 + \left(8 a + 11\right)\cdot 47 + \left(6 a + 42\right)\cdot 47^{2} + \left(26 a + 28\right)\cdot 47^{3} + \left(22 a + 4\right)\cdot 47^{4} + \left(28 a + 9\right)\cdot 47^{5} + \left(29 a + 37\right)\cdot 47^{6} + \left(36 a + 31\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 34 + \left(42 a + 8\right)\cdot 47 + \left(22 a + 36\right)\cdot 47^{2} + \left(11 a + 30\right)\cdot 47^{3} + \left(39 a + 16\right)\cdot 47^{4} + \left(6 a + 20\right)\cdot 47^{5} + \left(26 a + 26\right)\cdot 47^{6} + \left(42 a + 29\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 15 + \left(38 a + 15\right)\cdot 47 + \left(40 a + 46\right)\cdot 47^{2} + \left(20 a + 27\right)\cdot 47^{3} + \left(24 a + 23\right)\cdot 47^{4} + \left(18 a + 43\right)\cdot 47^{5} + \left(17 a + 20\right)\cdot 47^{6} + \left(10 a + 28\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 7\cdot 47 + 13\cdot 47^{2} + 12\cdot 47^{3} + 47^{4} + 24\cdot 47^{5} + 47^{6} + 25\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,6)$ |
| $(1,4)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,4,2)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,5,6,3)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
