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:
| $r_{ 1 }$ |
$=$ |
$ 13 + 32\cdot 47 + 33\cdot 47^{2} + 22\cdot 47^{3} + 11\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 a + 20 + \left(44 a + 31\right)\cdot 47 + \left(46 a + 5\right)\cdot 47^{2} + \left(45 a + 13\right)\cdot 47^{3} + \left(10 a + 6\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 12\cdot 47 + 21\cdot 47^{2} + 20\cdot 47^{3} + 43\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 a + 19 + \left(26 a + 12\right)\cdot 47 + \left(4 a + 45\right)\cdot 47^{2} + \left(12 a + 26\right)\cdot 47^{3} + \left(4 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 13 + \left(20 a + 22\right)\cdot 47 + \left(42 a + 27\right)\cdot 47^{2} + \left(34 a + 46\right)\cdot 47^{3} + \left(42 a + 46\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 14 + \left(2 a + 30\right)\cdot 47 + 7\cdot 47^{2} + \left(a + 11\right)\cdot 47^{3} + \left(36 a + 29\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(2,6)$ |
$-2$ |
| $9$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,6)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,5,6,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,2,5,6,3)$ |
$0$ |
| $12$ |
$6$ |
$(2,6)(3,4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
