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:
| $r_{ 1 }$ |
$=$ |
$ 37 a + 12 + 14 a\cdot 47 + \left(21 a + 42\right)\cdot 47^{2} + \left(16 a + 38\right)\cdot 47^{3} + \left(14 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 39 + \left(32 a + 39\right)\cdot 47 + \left(25 a + 22\right)\cdot 47^{2} + \left(30 a + 3\right)\cdot 47^{3} + \left(32 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 8 + \left(4 a + 34\right)\cdot 47 + \left(31 a + 4\right)\cdot 47^{2} + \left(26 a + 23\right)\cdot 47^{3} + \left(39 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 16 + \left(29 a + 40\right)\cdot 47 + \left(26 a + 15\right)\cdot 47^{2} + \left(26 a + 25\right)\cdot 47^{3} + \left(38 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 20 + \left(17 a + 2\right)\cdot 47 + \left(20 a + 40\right)\cdot 47^{2} + \left(20 a + 4\right)\cdot 47^{3} + \left(8 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 a + 46 + \left(42 a + 23\right)\cdot 47 + \left(15 a + 15\right)\cdot 47^{2} + \left(20 a + 45\right)\cdot 47^{3} + \left(7 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,4,6)(2,5,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,6,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,4,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
| $3$ | $6$ | $(1,3,6,5,4,2)$ | $0$ |
| $3$ | $6$ | $(1,2,4,5,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
