Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 + 8\cdot 47^{2} + 18\cdot 47^{3} + 14\cdot 47^{4} + 46\cdot 47^{5} + 18\cdot 47^{6} + 23\cdot 47^{7} + 3\cdot 47^{8} + 44\cdot 47^{9} +O\left(47^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 8 + \left(46 a + 29\right)\cdot 47 + \left(34 a + 2\right)\cdot 47^{2} + \left(27 a + 3\right)\cdot 47^{3} + \left(5 a + 31\right)\cdot 47^{4} + \left(30 a + 37\right)\cdot 47^{5} + \left(21 a + 25\right)\cdot 47^{6} + \left(9 a + 29\right)\cdot 47^{7} + \left(19 a + 35\right)\cdot 47^{8} + \left(34 a + 14\right)\cdot 47^{9} +O\left(47^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 5\cdot 47 + 23\cdot 47^{2} + 13\cdot 47^{3} + 13\cdot 47^{4} + 39\cdot 47^{5} + 30\cdot 47^{6} + 28\cdot 47^{7} + 24\cdot 47^{8} + 16\cdot 47^{9} +O\left(47^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 31 + 40\cdot 47 + \left(12 a + 25\right)\cdot 47^{2} + \left(19 a + 23\right)\cdot 47^{3} + \left(41 a + 14\right)\cdot 47^{4} + \left(16 a + 45\right)\cdot 47^{5} + \left(25 a + 38\right)\cdot 47^{6} + \left(37 a + 26\right)\cdot 47^{7} + \left(27 a + 17\right)\cdot 47^{8} + \left(12 a + 17\right)\cdot 47^{9} +O\left(47^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 a + 8 + \left(20 a + 11\right)\cdot 47 + \left(7 a + 20\right)\cdot 47^{2} + \left(36 a + 32\right)\cdot 47^{3} + \left(37 a + 37\right)\cdot 47^{4} + \left(34 a + 40\right)\cdot 47^{5} + 29 a\cdot 47^{6} + \left(35 a + 19\right)\cdot 47^{7} + \left(37 a + 33\right)\cdot 47^{8} + \left(12 a + 6\right)\cdot 47^{9} +O\left(47^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 6 + \left(26 a + 7\right)\cdot 47 + \left(39 a + 14\right)\cdot 47^{2} + \left(10 a + 3\right)\cdot 47^{3} + \left(9 a + 30\right)\cdot 47^{4} + \left(12 a + 25\right)\cdot 47^{5} + \left(17 a + 25\right)\cdot 47^{6} + \left(11 a + 13\right)\cdot 47^{7} + \left(9 a + 26\right)\cdot 47^{8} + \left(34 a + 41\right)\cdot 47^{9} +O\left(47^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(1,4)(3,5)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(4,5)$ | $-1$ |
| $6$ | $2$ | $(2,4)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,4,2)(3,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,3,4)$ | $1$ |
| $6$ | $4$ | $(1,3)(2,5,6,4)$ | $-1$ |
| $8$ | $6$ | $(1,5,6,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
