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