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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 21 + \left(2 a + 34\right)\cdot 47 + \left(32 a + 1\right)\cdot 47^{2} + \left(20 a + 21\right)\cdot 47^{3} + \left(7 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 36 + \left(5 a + 43\right)\cdot 47 + \left(37 a + 35\right)\cdot 47^{2} + 43 a\cdot 47^{3} + \left(23 a + 9\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 4 + \left(44 a + 24\right)\cdot 47 + \left(14 a + 16\right)\cdot 47^{2} + \left(26 a + 30\right)\cdot 47^{3} + \left(39 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 32 + \left(41 a + 10\right)\cdot 47 + \left(9 a + 10\right)\cdot 47^{2} + \left(3 a + 4\right)\cdot 47^{3} + \left(23 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 36\cdot 47 + 40\cdot 47^{2} + 11\cdot 47^{3} + 28\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 38\cdot 47 + 35\cdot 47^{2} + 25\cdot 47^{3} + 12\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$2$ |
| $6$ |
$2$ |
$(3,5)$ |
$0$ |
| $9$ |
$2$ |
$(3,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,5)$ |
$-2$ |
| $4$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,2)(3,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,3,6,5,2)$ |
$-1$ |
| $12$ |
$6$ |
$(2,4,6)(3,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
