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:
| $r_{ 1 }$ |
$=$ |
$ 45 a + 25 + \left(3 a + 23\right)\cdot 47 + \left(29 a + 13\right)\cdot 47^{2} + \left(7 a + 1\right)\cdot 47^{3} + \left(24 a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 8\cdot 47 + 45\cdot 47^{2} + 9\cdot 47^{3} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 21 + \left(43 a + 33\right)\cdot 47 + \left(17 a + 20\right)\cdot 47^{2} + \left(39 a + 34\right)\cdot 47^{3} + \left(22 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 38\cdot 47 + 20\cdot 47^{2} + 41\cdot 47^{3} + 12\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 7 + \left(18 a + 4\right)\cdot 47 + \left(39 a + 37\right)\cdot 47^{2} + \left(29 a + 16\right)\cdot 47^{3} + \left(44 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 21 + \left(28 a + 33\right)\cdot 47 + \left(7 a + 3\right)\cdot 47^{2} + \left(17 a + 37\right)\cdot 47^{3} + \left(2 a + 24\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 value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
