Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 a + 3 + \left(25 a + 23\right)\cdot 47 + \left(13 a + 35\right)\cdot 47^{2} + \left(10 a + 15\right)\cdot 47^{3} + 23\cdot 47^{4} + \left(25 a + 33\right)\cdot 47^{5} + \left(44 a + 4\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 38 + \left(21 a + 33\right)\cdot 47 + \left(33 a + 36\right)\cdot 47^{2} + \left(36 a + 22\right)\cdot 47^{3} + \left(46 a + 13\right)\cdot 47^{4} + \left(21 a + 36\right)\cdot 47^{5} + \left(2 a + 21\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 4 + \left(46 a + 20\right)\cdot 47 + \left(32 a + 4\right)\cdot 47^{2} + \left(26 a + 19\right)\cdot 47^{3} + \left(29 a + 24\right)\cdot 47^{4} + \left(31 a + 34\right)\cdot 47^{5} + \left(17 a + 40\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 5 + \left(25 a + 18\right)\cdot 47 + \left(46 a + 33\right)\cdot 47^{2} + \left(36 a + 31\right)\cdot 47^{3} + \left(29 a + 23\right)\cdot 47^{4} + \left(9 a + 36\right)\cdot 47^{5} + \left(15 a + 27\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 5 + 42\cdot 47 + \left(14 a + 23\right)\cdot 47^{2} + \left(20 a + 39\right)\cdot 47^{3} + \left(17 a + 9\right)\cdot 47^{4} + \left(15 a + 21\right)\cdot 47^{5} + \left(29 a + 44\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 41 + \left(21 a + 3\right)\cdot 47 + 7\cdot 47^{2} + \left(10 a + 12\right)\cdot 47^{3} + \left(17 a + 46\right)\cdot 47^{4} + \left(37 a + 25\right)\cdot 47^{5} + \left(31 a + 1\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,6)$ |
| $(2,4,5)$ |
| $(1,5,6,2,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,3,6)(2,4,5)$ | $-1$ |
| $3$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
| $3$ | $6$ | $(1,4,3,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
