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:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 5 + \left(6 a + 25\right)\cdot 47 + \left(22 a + 8\right)\cdot 47^{2} + \left(11 a + 8\right)\cdot 47^{3} + \left(41 a + 40\right)\cdot 47^{4} + \left(23 a + 36\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 + 24\cdot 47 + 2\cdot 47^{2} + 5\cdot 47^{3} + 18\cdot 47^{4} + 6\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a + 17 + \left(40 a + 31\right)\cdot 47 + \left(24 a + 46\right)\cdot 47^{2} + \left(35 a + 8\right)\cdot 47^{3} + \left(5 a + 17\right)\cdot 47^{4} + \left(23 a + 43\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 22 + \left(34 a + 14\right)\cdot 47 + \left(17 a + 23\right)\cdot 47^{2} + \left(5 a + 25\right)\cdot 47^{3} + \left(39 a + 4\right)\cdot 47^{4} + \left(35 a + 7\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 11\cdot 47 + 35\cdot 47^{2} + 27\cdot 47^{3} + 30\cdot 47^{4} + 7\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 a + 26 + \left(12 a + 33\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(41 a + 18\right)\cdot 47^{3} + \left(7 a + 30\right)\cdot 47^{4} + \left(11 a + 39\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
