Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 36 + \left(46 a + 32\right)\cdot 47 + \left(32 a + 12\right)\cdot 47^{2} + \left(5 a + 4\right)\cdot 47^{3} + \left(16 a + 27\right)\cdot 47^{4} + \left(46 a + 30\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 37\cdot 47 + 32\cdot 47^{2} + 29\cdot 47^{3} + 46\cdot 47^{4} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 31\cdot 47 + 33\cdot 47^{2} + 8\cdot 47^{3} + 25\cdot 47^{4} + 16\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 7 + 22\cdot 47 + \left(14 a + 32\right)\cdot 47^{2} + \left(41 a + 29\right)\cdot 47^{3} + \left(30 a + 6\right)\cdot 47^{4} + 13\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 8 + \left(2 a + 13\right)\cdot 47 + \left(11 a + 28\right)\cdot 47^{2} + \left(32 a + 7\right)\cdot 47^{3} + \left(46 a + 34\right)\cdot 47^{4} + \left(40 a + 45\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 36 + \left(44 a + 3\right)\cdot 47 + \left(35 a + 1\right)\cdot 47^{2} + \left(14 a + 14\right)\cdot 47^{3} + 47^{4} + \left(6 a + 34\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(4,6)$ |
| $(1,4)$ |
| $(1,5,2)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,4)$ |
$1$ |
| $3$ |
$2$ |
$(1,4)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(2,5)(3,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,5,2)(3,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,4,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,4)(2,6,3,5)$ |
$-1$ |
| $8$ |
$6$ |
$(1,6,3,4,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
