Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 15 + \left(a + 14\right)\cdot 47 + \left(2 a + 22\right)\cdot 47^{2} + \left(23 a + 1\right)\cdot 47^{3} + \left(28 a + 30\right)\cdot 47^{4} + \left(24 a + 36\right)\cdot 47^{5} + \left(26 a + 32\right)\cdot 47^{6} + \left(42 a + 17\right)\cdot 47^{7} + \left(10 a + 10\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 8\cdot 47 + 23\cdot 47^{2} + 2\cdot 47^{3} + 42\cdot 47^{5} + 25\cdot 47^{6} + 13\cdot 47^{7} + 39\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 29 + \left(18 a + 37\right)\cdot 47 + \left(5 a + 3\right)\cdot 47^{2} + \left(35 a + 38\right)\cdot 47^{3} + \left(2 a + 14\right)\cdot 47^{4} + \left(36 a + 12\right)\cdot 47^{5} + \left(36 a + 28\right)\cdot 47^{6} + 17 a\cdot 47^{7} + \left(13 a + 19\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 32 + \left(45 a + 32\right)\cdot 47 + \left(44 a + 24\right)\cdot 47^{2} + \left(23 a + 45\right)\cdot 47^{3} + \left(18 a + 16\right)\cdot 47^{4} + \left(22 a + 10\right)\cdot 47^{5} + \left(20 a + 14\right)\cdot 47^{6} + \left(4 a + 29\right)\cdot 47^{7} + \left(36 a + 36\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 38\cdot 47 + 23\cdot 47^{2} + 44\cdot 47^{3} + 46\cdot 47^{4} + 4\cdot 47^{5} + 21\cdot 47^{6} + 33\cdot 47^{7} + 7\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 18 + \left(28 a + 9\right)\cdot 47 + \left(41 a + 43\right)\cdot 47^{2} + \left(11 a + 8\right)\cdot 47^{3} + \left(44 a + 32\right)\cdot 47^{4} + \left(10 a + 34\right)\cdot 47^{5} + \left(10 a + 18\right)\cdot 47^{6} + \left(29 a + 46\right)\cdot 47^{7} + \left(33 a + 27\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(2,3,5,6)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $-1$ |
| $8$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
