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:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 29 + \left(29 a + 4\right)\cdot 47 + \left(12 a + 2\right)\cdot 47^{2} + \left(21 a + 23\right)\cdot 47^{3} + \left(29 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 33\cdot 47 + 4\cdot 47^{2} + 34\cdot 47^{3} + 24\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 14 + 17 a\cdot 47 + \left(34 a + 45\right)\cdot 47^{2} + \left(25 a + 5\right)\cdot 47^{3} + \left(17 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 41 + \left(10 a + 26\right)\cdot 47 + \left(42 a + 1\right)\cdot 47^{2} + \left(13 a + 20\right)\cdot 47^{3} + \left(23 a + 45\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 12 + \left(36 a + 38\right)\cdot 47 + \left(4 a + 28\right)\cdot 47^{2} + \left(33 a + 5\right)\cdot 47^{3} + \left(23 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 + 37\cdot 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\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 |
| $1$ | $1$ | $()$ | $10$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
