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:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 46 + \left(37 a + 6\right)\cdot 47 + \left(27 a + 10\right)\cdot 47^{2} + \left(41 a + 12\right)\cdot 47^{3} + \left(4 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 16\cdot 47 + 34\cdot 47^{2} + 8\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 23\cdot 47 + 16\cdot 47^{2} + 30\cdot 47^{3} + 27\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 23 + \left(9 a + 22\right)\cdot 47 + \left(19 a + 28\right)\cdot 47^{2} + \left(5 a + 20\right)\cdot 47^{3} + \left(42 a + 27\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + \left(19 a + 15\right)\cdot 47 + \left(18 a + 17\right)\cdot 47^{2} + \left(29 a + 14\right)\cdot 47^{3} + \left(13 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 43 + \left(27 a + 9\right)\cdot 47 + \left(28 a + 34\right)\cdot 47^{2} + \left(17 a + 7\right)\cdot 47^{3} + \left(33 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,2)(3,4,5)$ |
| $(2,3)(5,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $8$ | $3$ | $(1,6,2)(3,4,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
