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 + 33\cdot 47 + 33\cdot 47^{2} + 17\cdot 47^{3} + 40\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 40 + \left(33 a + 2\right)\cdot 47 + \left(27 a + 26\right)\cdot 47^{2} + \left(38 a + 6\right)\cdot 47^{3} + \left(36 a + 36\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 41 + \left(44 a + 22\right)\cdot 47 + \left(36 a + 45\right)\cdot 47^{2} + \left(41 a + 4\right)\cdot 47^{3} + \left(3 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 a + 1 + \left(13 a + 18\right)\cdot 47 + \left(19 a + 1\right)\cdot 47^{2} + \left(8 a + 9\right)\cdot 47^{3} + \left(10 a + 24\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 28 + 2 a\cdot 47 + \left(10 a + 28\right)\cdot 47^{2} + \left(5 a + 4\right)\cdot 47^{3} + \left(43 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 15\cdot 47 + 6\cdot 47^{2} + 4\cdot 47^{3} + 18\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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
