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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 40 + \left(31 a + 38\right)\cdot 47 + \left(36 a + 45\right)\cdot 47^{2} + \left(15 a + 31\right)\cdot 47^{3} + \left(7 a + 16\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 37 + \left(15 a + 32\right)\cdot 47 + \left(10 a + 40\right)\cdot 47^{2} + \left(31 a + 26\right)\cdot 47^{3} + \left(39 a + 15\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 21 + \left(15 a + 44\right)\cdot 47 + \left(10 a + 40\right)\cdot 47^{2} + \left(31 a + 46\right)\cdot 47^{3} + \left(39 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 35 + \left(31 a + 13\right)\cdot 47 + \left(36 a + 33\right)\cdot 47^{2} + \left(15 a + 17\right)\cdot 47^{3} + \left(7 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 a + 32 + \left(15 a + 7\right)\cdot 47 + \left(10 a + 28\right)\cdot 47^{2} + \left(31 a + 12\right)\cdot 47^{3} + \left(39 a + 11\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 24 + \left(31 a + 3\right)\cdot 47 + \left(36 a + 46\right)\cdot 47^{2} + \left(15 a + 4\right)\cdot 47^{3} + \left(7 a + 43\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6,2,4,3)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,2,4,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,4,2,6,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
