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