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:
| $r_{ 1 }$ |
$=$ |
$ 1 + 44\cdot 47 + 3\cdot 47^{2} + 27\cdot 47^{3} + 20\cdot 47^{4} + 5\cdot 47^{5} + 41\cdot 47^{6} + 33\cdot 47^{7} + 14\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 33 + \left(22 a + 37\right)\cdot 47 + \left(21 a + 34\right)\cdot 47^{2} + \left(10 a + 37\right)\cdot 47^{3} + \left(36 a + 23\right)\cdot 47^{4} + \left(35 a + 37\right)\cdot 47^{5} + \left(7 a + 27\right)\cdot 47^{6} + \left(31 a + 43\right)\cdot 47^{7} + \left(8 a + 30\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 43 + \left(16 a + 4\right)\cdot 47 + \left(13 a + 39\right)\cdot 47^{2} + \left(18 a + 28\right)\cdot 47^{3} + \left(28 a + 18\right)\cdot 47^{4} + \left(23 a + 38\right)\cdot 47^{5} + 32\cdot 47^{6} + \left(31 a + 38\right)\cdot 47^{7} + \left(10 a + 37\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 42 + \left(30 a + 14\right)\cdot 47 + \left(33 a + 2\right)\cdot 47^{2} + \left(28 a + 5\right)\cdot 47^{3} + \left(18 a + 10\right)\cdot 47^{4} + \left(23 a + 10\right)\cdot 47^{5} + \left(46 a + 10\right)\cdot 47^{6} + \left(15 a + 6\right)\cdot 47^{7} + \left(36 a + 28\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 34\cdot 47 + 5\cdot 47^{2} + 5\cdot 47^{3} + 29\cdot 47^{4} + 23\cdot 47^{5} + 21\cdot 47^{6} + 14\cdot 47^{7} + 12\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 1 + \left(24 a + 5\right)\cdot 47 + \left(25 a + 8\right)\cdot 47^{2} + \left(36 a + 37\right)\cdot 47^{3} + \left(10 a + 38\right)\cdot 47^{4} + \left(11 a + 25\right)\cdot 47^{5} + \left(39 a + 7\right)\cdot 47^{6} + \left(15 a + 4\right)\cdot 47^{7} + \left(38 a + 17\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,6,5)$ |
| $(1,2)(3,5)$ |
| $(1,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,5)(2,3)$ | $-1$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,5)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,4)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(1,2,5,3)$ | $1$ |
| $6$ | $4$ | $(1,6,5,4)(2,3)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,5,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
