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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + \left(24 a + 44\right)\cdot 47 + \left(6 a + 7\right)\cdot 47^{2} + \left(27 a + 32\right)\cdot 47^{3} + 24\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 10 + \left(22 a + 40\right)\cdot 47 + \left(40 a + 43\right)\cdot 47^{2} + \left(19 a + 32\right)\cdot 47^{3} + \left(46 a + 45\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 6 + \left(24 a + 37\right)\cdot 47 + \left(28 a + 10\right)\cdot 47^{2} + \left(14 a + 32\right)\cdot 47^{3} + \left(7 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 30\cdot 47 + 11\cdot 47^{2} + 16\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 9 + \left(22 a + 14\right)\cdot 47 + \left(18 a + 43\right)\cdot 47^{2} + \left(32 a + 32\right)\cdot 47^{3} + \left(39 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 21\cdot 47 + 23\cdot 47^{2} + 41\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
