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:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 46 + \left(15 a + 44\right)\cdot 47 + \left(26 a + 17\right)\cdot 47^{2} + \left(45 a + 45\right)\cdot 47^{3} + \left(39 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 19 + \left(43 a + 7\right)\cdot 47 + \left(31 a + 22\right)\cdot 47^{2} + \left(21 a + 12\right)\cdot 47^{3} + \left(46 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 43 + \left(3 a + 34\right)\cdot 47 + \left(15 a + 42\right)\cdot 47^{2} + \left(25 a + 23\right)\cdot 47^{3} + 18\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 14\cdot 47 + 10\cdot 47^{2} + 17\cdot 47^{3} + 5\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 4\cdot 47 + 40\cdot 47^{2} + 25\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 38 + \left(31 a + 33\right)\cdot 47 + \left(20 a + 7\right)\cdot 47^{2} + \left(a + 16\right)\cdot 47^{3} + \left(7 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$10$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $15$ |
$2$ |
$(1,2)$ |
$2$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
