Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 a + 25 + \left(9 a + 29\right)\cdot 47 + \left(19 a + 5\right)\cdot 47^{2} + \left(19 a + 29\right)\cdot 47^{3} + \left(7 a + 8\right)\cdot 47^{4} + \left(29 a + 9\right)\cdot 47^{5} + \left(36 a + 45\right)\cdot 47^{6} + \left(46 a + 32\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 44 + \left(37 a + 15\right)\cdot 47 + \left(27 a + 34\right)\cdot 47^{2} + \left(27 a + 1\right)\cdot 47^{3} + \left(39 a + 4\right)\cdot 47^{4} + \left(17 a + 13\right)\cdot 47^{5} + \left(10 a + 42\right)\cdot 47^{6} + 42\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 7 + \left(34 a + 24\right)\cdot 47 + \left(45 a + 5\right)\cdot 47^{2} + \left(17 a + 15\right)\cdot 47^{3} + \left(6 a + 12\right)\cdot 47^{4} + \left(41 a + 28\right)\cdot 47^{5} + \left(23 a + 46\right)\cdot 47^{6} + \left(9 a + 4\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 29\cdot 47 + 37\cdot 47^{2} + 47^{3} + 25\cdot 47^{4} + 28\cdot 47^{5} + 45\cdot 47^{6} + 14\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 + 18\cdot 47 + 42\cdot 47^{2} + 40\cdot 47^{3} + 36\cdot 47^{4} + 4\cdot 47^{5} + 2\cdot 47^{6} + 45\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 6 + \left(12 a + 23\right)\cdot 47 + \left(a + 15\right)\cdot 47^{2} + \left(29 a + 5\right)\cdot 47^{3} + \left(40 a + 7\right)\cdot 47^{4} + \left(5 a + 10\right)\cdot 47^{5} + \left(23 a + 6\right)\cdot 47^{6} + 37 a\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5)$ |
| $(3,4)(5,6)$ |
| $(1,4,3)(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)(4,5)$ |
$-1$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $6$ |
$2$ |
$(3,4)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,4,3)(2,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(3,5,6,4)$ |
$1$ |
| $6$ |
$4$ |
$(1,2)(3,5,6,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,3,5,2,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
