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:
| $r_{ 1 }$ |
$=$ |
$ 23 + 24\cdot 47 + 25\cdot 47^{2} + 32\cdot 47^{3} + 46\cdot 47^{4} + 25\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 36 + 24\cdot 47 + \left(23 a + 35\right)\cdot 47^{2} + \left(26 a + 32\right)\cdot 47^{3} + \left(46 a + 33\right)\cdot 47^{4} + \left(26 a + 15\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a + 43 + \left(23 a + 30\right)\cdot 47 + \left(32 a + 13\right)\cdot 47^{2} + \left(40 a + 6\right)\cdot 47^{3} + \left(20 a + 23\right)\cdot 47^{4} + \left(42 a + 25\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 4\cdot 47 + 24\cdot 47^{2} + 45\cdot 47^{3} + 6\cdot 47^{4} + 8\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 3 + \left(46 a + 18\right)\cdot 47 + \left(23 a + 34\right)\cdot 47^{2} + \left(20 a + 15\right)\cdot 47^{3} + 6\cdot 47^{4} + \left(20 a + 23\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 29 + \left(23 a + 38\right)\cdot 47 + \left(14 a + 7\right)\cdot 47^{2} + \left(6 a + 8\right)\cdot 47^{3} + \left(26 a + 24\right)\cdot 47^{4} + \left(4 a + 42\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,2)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,6,3)(2,5,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,6,4,3,2)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
