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:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 7 + \left(14 a + 5\right)\cdot 47 + \left(42 a + 12\right)\cdot 47^{2} + \left(10 a + 10\right)\cdot 47^{3} + \left(20 a + 32\right)\cdot 47^{4} + \left(17 a + 39\right)\cdot 47^{5} + \left(44 a + 34\right)\cdot 47^{6} + \left(17 a + 27\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 29 + \left(36 a + 19\right)\cdot 47 + \left(30 a + 34\right)\cdot 47^{2} + \left(36 a + 25\right)\cdot 47^{3} + \left(38 a + 26\right)\cdot 47^{4} + \left(44 a + 21\right)\cdot 47^{5} + \left(5 a + 16\right)\cdot 47^{6} + \left(16 a + 10\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 14\cdot 47 + 36\cdot 47^{2} + 46\cdot 47^{3} + 5\cdot 47^{4} + 41\cdot 47^{5} + 13\cdot 47^{6} + 28\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 40 + \left(32 a + 41\right)\cdot 47 + \left(4 a + 34\right)\cdot 47^{2} + \left(36 a + 36\right)\cdot 47^{3} + \left(26 a + 14\right)\cdot 47^{4} + \left(29 a + 7\right)\cdot 47^{5} + \left(2 a + 12\right)\cdot 47^{6} + \left(29 a + 19\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 18 + \left(10 a + 27\right)\cdot 47 + \left(16 a + 12\right)\cdot 47^{2} + \left(10 a + 21\right)\cdot 47^{3} + \left(8 a + 20\right)\cdot 47^{4} + \left(2 a + 25\right)\cdot 47^{5} + \left(41 a + 30\right)\cdot 47^{6} + \left(30 a + 36\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 32\cdot 47 + 10\cdot 47^{2} + 41\cdot 47^{4} + 5\cdot 47^{5} + 33\cdot 47^{6} + 18\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,6,4,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,4,3)(2,5)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
