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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 34 + 25\cdot 47 + \left(16 a + 23\right)\cdot 47^{2} + \left(22 a + 22\right)\cdot 47^{3} + \left(25 a + 30\right)\cdot 47^{4} + \left(25 a + 37\right)\cdot 47^{5} + \left(12 a + 17\right)\cdot 47^{6} + \left(35 a + 15\right)\cdot 47^{7} + 34\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 31 + \left(46 a + 4\right)\cdot 47 + \left(30 a + 8\right)\cdot 47^{2} + \left(24 a + 4\right)\cdot 47^{3} + \left(21 a + 12\right)\cdot 47^{4} + \left(21 a + 16\right)\cdot 47^{5} + \left(34 a + 17\right)\cdot 47^{6} + \left(11 a + 26\right)\cdot 47^{7} + 46 a\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 15 + \left(12 a + 14\right)\cdot 47 + \left(24 a + 38\right)\cdot 47^{2} + 19\cdot 47^{3} + \left(12 a + 35\right)\cdot 47^{4} + \left(13 a + 10\right)\cdot 47^{5} + \left(34 a + 34\right)\cdot 47^{6} + \left(24 a + 31\right)\cdot 47^{7} + \left(41 a + 22\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 5\cdot 47 + 28\cdot 47^{2} + 29\cdot 47^{3} + 10\cdot 47^{4} + 43\cdot 47^{5} + 21\cdot 47^{6} + 9\cdot 47^{7} + 28\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 18\cdot 47 + 15\cdot 47^{2} + 21\cdot 47^{3} + 40\cdot 47^{4} + 7\cdot 47^{5} + 7\cdot 47^{6} + 11\cdot 47^{7} + 21\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 43 + \left(34 a + 24\right)\cdot 47 + \left(22 a + 27\right)\cdot 47^{2} + \left(46 a + 43\right)\cdot 47^{3} + \left(34 a + 11\right)\cdot 47^{4} + \left(33 a + 25\right)\cdot 47^{5} + \left(12 a + 42\right)\cdot 47^{6} + \left(22 a + 46\right)\cdot 47^{7} + \left(5 a + 33\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,6,5)$ |
| $(1,4)(2,5)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(3,6)(4,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $1$ |
| $8$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $6$ | $4$ | $(3,5,6,4)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,5,6,4)$ | $1$ |
| $8$ | $6$ | $(1,3,5,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
