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:
| $r_{ 1 }$ |
$=$ |
$ 5 + 22\cdot 47 + 38\cdot 47^{2} + 22\cdot 47^{3} + 42\cdot 47^{4} + 12\cdot 47^{5} + 41\cdot 47^{6} + 42\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 36 + \left(44 a + 19\right)\cdot 47 + \left(17 a + 8\right)\cdot 47^{2} + \left(39 a + 5\right)\cdot 47^{3} + \left(35 a + 33\right)\cdot 47^{4} + \left(34 a + 23\right)\cdot 47^{5} + \left(46 a + 20\right)\cdot 47^{6} + \left(26 a + 45\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 44 + 20 a\cdot 47 + \left(31 a + 26\right)\cdot 47^{2} + \left(41 a + 6\right)\cdot 47^{3} + \left(30 a + 27\right)\cdot 47^{4} + \left(5 a + 43\right)\cdot 47^{5} + \left(45 a + 46\right)\cdot 47^{6} + \left(25 a + 32\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 7 + \left(2 a + 5\right)\cdot 47 + 29 a\cdot 47^{2} + \left(7 a + 19\right)\cdot 47^{3} + \left(11 a + 18\right)\cdot 47^{4} + \left(12 a + 10\right)\cdot 47^{5} + 32\cdot 47^{6} + \left(20 a + 5\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 19 + \left(26 a + 30\right)\cdot 47 + \left(15 a + 21\right)\cdot 47^{2} + \left(5 a + 11\right)\cdot 47^{3} + 16 a\cdot 47^{4} + \left(41 a + 24\right)\cdot 47^{5} + \left(a + 37\right)\cdot 47^{6} + \left(21 a + 39\right)\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 15\cdot 47 + 46\cdot 47^{2} + 28\cdot 47^{3} + 19\cdot 47^{4} + 26\cdot 47^{5} + 9\cdot 47^{6} + 21\cdot 47^{7} +O\left(47^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)$ |
| $(1,3)(2,6)(4,5)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(3,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,3)(2,6)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,4,2)(3,6,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,2,6,4,3)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
