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:
| $r_{ 1 }$ |
$=$ |
$ 40 a^{2} + 25 a + 28 + \left(27 a^{2} + 28 a + 7\right)\cdot 47 + \left(6 a^{2} + 8 a + 26\right)\cdot 47^{2} + \left(16 a^{2} + 10 a + 18\right)\cdot 47^{3} + \left(22 a^{2} + 41 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a^{2} + 36 a + 17 + \left(12 a^{2} + 22 a + 23\right)\cdot 47 + \left(20 a^{2} + 4 a + 6\right)\cdot 47^{2} + \left(11 a^{2} + 30 a + 9\right)\cdot 47^{3} + \left(11 a^{2} + 30 a + 11\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a^{2} + 4 a + 40 + \left(10 a^{2} + 42 a + 29\right)\cdot 47 + \left(18 a^{2} + 19 a + 10\right)\cdot 47^{2} + \left(32 a^{2} + 21 a + 36\right)\cdot 47^{3} + \left(46 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a^{2} + 39 a + 27 + \left(43 a^{2} + 19 a + 2\right)\cdot 47 + \left(7 a^{2} + 46 a + 37\right)\cdot 47^{2} + \left(42 a^{2} + 29 a + 8\right)\cdot 47^{3} + \left(16 a^{2} + 11 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a^{2} + 33 a + 34 + \left(6 a^{2} + 42 a + 12\right)\cdot 47 + \left(20 a^{2} + 33 a + 6\right)\cdot 47^{2} + \left(19 a^{2} + 6 a + 25\right)\cdot 47^{3} + \left(13 a^{2} + 22 a + 15\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a^{2} + 34 a + 36 + \left(25 a + 8\right)\cdot 47 + \left(7 a^{2} + 20 a + 11\right)\cdot 47^{2} + \left(6 a^{2} + 29 a + 23\right)\cdot 47^{3} + \left(14 a^{2} + 9 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 a^{2} + 4 a + 35 + \left(40 a^{2} + 32 a + 43\right)\cdot 47 + \left(20 a^{2} + 27 a + 15\right)\cdot 47^{2} + \left(19 a^{2} + 42 a + 10\right)\cdot 47^{3} + \left(29 a^{2} + 35 a + 43\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 11 a^{2} + 19 a + 25 + \left(16 a^{2} + 5 a + 39\right)\cdot 47 + \left(14 a^{2} + 12 a + 25\right)\cdot 47^{2} + \left(24 a^{2} + 29 a + 12\right)\cdot 47^{3} + \left(42 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 43 a^{2} + 41 a + 42 + \left(29 a^{2} + 15 a + 19\right)\cdot 47 + \left(25 a^{2} + 14 a + 1\right)\cdot 47^{2} + \left(16 a^{2} + 35 a + 44\right)\cdot 47^{3} + \left(32 a^{2} + 41 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2)(3,9)(4,6)(7,8)$ |
| $(1,2,5)(3,4,7)(6,9,8)$ |
| $(1,8,7,2,6,3,5,9,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,2)(3,9)(4,6)(7,8)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,2,5)(3,4,7)(6,9,8)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,8,7,2,6,3,5,9,4)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,7,6,5,4,8,2,3,9)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,6,4,2,9,7,5,8,3)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
