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:
| $r_{ 1 }$ |
$=$ |
$ 39 a^{2} + 39 a + 33 + \left(38 a^{2} + 7 a + 13\right)\cdot 47 + \left(18 a^{2} + 30 a + 5\right)\cdot 47^{2} + \left(17 a^{2} + 24 a + 23\right)\cdot 47^{3} + \left(34 a^{2} + 29 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a^{2} + 12 a + 21 + \left(18 a^{2} + 46 a + 1\right)\cdot 47 + \left(29 a^{2} + 29 a + 22\right)\cdot 47^{2} + \left(24 a^{2} + 38 a + 31\right)\cdot 47^{3} + \left(33 a^{2} + 9 a + 11\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a^{2} + 26 a + 25 + \left(33 a^{2} + 2 a + 41\right)\cdot 47 + \left(13 a^{2} + 18 a + 33\right)\cdot 47^{2} + \left(23 a^{2} + 26 a + 44\right)\cdot 47^{3} + \left(39 a^{2} + 15 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 a^{2} + a + 15 + \left(41 a^{2} + 4 a\right)\cdot 47 + \left(27 a^{2} + 2 a + 19\right)\cdot 47^{2} + \left(15 a^{2} + 32 a + 13\right)\cdot 47^{3} + \left(11 a^{2} + 24 a + 14\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a^{2} + 26 a + 34 + \left(11 a^{2} + 4 a + 45\right)\cdot 47 + \left(9 a^{2} + 45 a + 24\right)\cdot 47^{2} + \left(a^{2} + 45 a\right)\cdot 47^{3} + \left(2 a + 17\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a^{2} + 20 a + 3 + \left(20 a^{2} + 46 a + 24\right)\cdot 47 + \left(41 a^{2} + 20 a + 3\right)\cdot 47^{2} + \left(19 a + 37\right)\cdot 47^{3} + \left(28 a^{2} + 9 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 36 a^{2} + 42 a + 20 + \left(a^{2} + 39 a + 25\right)\cdot 47 + \left(24 a^{2} + 30 a + 7\right)\cdot 47^{2} + \left(22 a^{2} + 21 a + 43\right)\cdot 47^{3} + \left(7 a^{2} + 28 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a^{2} + 34 a + 22 + \left(33 a^{2} + 43 a + 30\right)\cdot 47 + \left(36 a^{2} + 14 a + 36\right)\cdot 47^{2} + \left(6 a^{2} + 23 a + 42\right)\cdot 47^{3} + \left(2 a^{2} + 12 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 31 a^{2} + 35 a + 17 + \left(34 a^{2} + 39 a + 5\right)\cdot 47 + \left(33 a^{2} + 42 a + 35\right)\cdot 47^{2} + \left(28 a^{2} + 2 a + 45\right)\cdot 47^{3} + \left(31 a^{2} + 8 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5)(2,4)(3,9)(6,7)$ |
| $(1,8,5,6,2,3,9,4,7)$ |
| $(1,6,9)(2,4,8)(3,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $9$ | $2$ | $(1,5)(2,4)(3,9)(6,7)$ | $0$ |
| $2$ | $3$ | $(1,6,9)(2,4,8)(3,7,5)$ | $-1$ |
| $2$ | $9$ | $(1,8,5,6,2,3,9,4,7)$ | $\zeta_{9}^{5} + \zeta_{9}^{4}$ |
| $2$ | $9$ | $(1,5,2,9,7,8,6,3,4)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ | $9$ | $(1,2,7,6,4,5,9,8,3)$ | $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
The blue line marks the conjugacy class containing complex conjugation.
