Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{3} + 3 x + 42 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 30 a + 2 + \left(37 a^{2} + 30 a + 27\right)\cdot 47 + \left(8 a^{2} + 21 a + 17\right)\cdot 47^{2} + \left(18 a^{2} + 24 a + 36\right)\cdot 47^{3} + \left(40 a^{2} + 45 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a^{2} + 43 a + 32 + \left(10 a^{2} + 11 a + 20\right)\cdot 47 + \left(33 a^{2} + 32 a + 19\right)\cdot 47^{2} + \left(46 a^{2} + 9 a + 46\right)\cdot 47^{3} + \left(10 a^{2} + 7 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 a^{2} + 12 a + 41 + \left(25 a^{2} + 5 a + 4\right)\cdot 47 + \left(11 a^{2} + 29 a + 23\right)\cdot 47^{2} + \left(31 a^{2} + 37 a + 15\right)\cdot 47^{3} + \left(42 a^{2} + 23 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a^{2} + 15 a + 17 + \left(5 a^{2} + 19 a + 11\right)\cdot 47 + \left(43 a^{2} + 34 a + 39\right)\cdot 47^{2} + \left(21 a^{2} + 22 a + 43\right)\cdot 47^{3} + \left(40 a^{2} + 25 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a^{2} + 16 a + 10 + \left(30 a^{2} + 5 a + 13\right)\cdot 47 + \left(16 a^{2} + 27 a + 33\right)\cdot 47^{2} + \left(26 a^{2} + 23 a + 5\right)\cdot 47^{3} + \left(21 a^{2} + 36 a + 43\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a^{2} + 2 a + 28 + \left(4 a^{2} + 44 a + 8\right)\cdot 47 + \left(42 a^{2} + 37 a + 37\right)\cdot 47^{2} + \left(6 a^{2} + 46 a + 13\right)\cdot 47^{3} + \left(13 a^{2} + 22 a + 26\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 19 a^{2} + 20 a + 38 + \left(14 a^{2} + 13 a + 28\right)\cdot 47 + \left(26 a^{2} + 3 a + 5\right)\cdot 47^{2} + \left(13 a^{2} + 36 a + 27\right)\cdot 47^{3} + \left(24 a^{2} + 8 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 23 a^{2} + 11 a + 46 + \left(2 a^{2} + 28 a + 4\right)\cdot 47 + \left(4 a^{2} + 16 a + 8\right)\cdot 47^{2} + \left(7 a^{2} + 34 a + 14\right)\cdot 47^{3} + \left(a^{2} + a + 2\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 34 a^{2} + 39 a + 21 + \left(10 a^{2} + 29 a + 21\right)\cdot 47 + \left(2 a^{2} + 32 a + 4\right)\cdot 47^{2} + \left(16 a^{2} + 46 a + 32\right)\cdot 47^{3} + \left(40 a^{2} + 15 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,2,4,8,9,6,7,3)$ |
| $(1,8)(3,9)(4,5)(6,7)$ |
| $(1,4,6)(2,9,3)(5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,8)(3,9)(4,5)(6,7)$ | $0$ |
| $2$ | $3$ | $(1,4,6)(2,9,3)(5,8,7)$ | $-1$ |
| $2$ | $9$ | $(1,5,2,4,8,9,6,7,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ | $9$ | $(1,2,8,6,3,5,4,9,7)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,8,3,4,7,2,6,5,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
