Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 30 + \left(18 a + 31\right)\cdot 47 + \left(2 a + 30\right)\cdot 47^{2} + \left(14 a + 4\right)\cdot 47^{3} + \left(6 a + 22\right)\cdot 47^{4} + \left(34 a + 16\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 35 + \left(16 a + 43\right)\cdot 47 + \left(43 a + 42\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(18 a + 16\right)\cdot 47^{4} + \left(a + 21\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 12\cdot 47 + 22\cdot 47^{2} + 20\cdot 47^{3} + 13\cdot 47^{4} + 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 12\cdot 47 + 8\cdot 47^{2} + 44\cdot 47^{4} + 17\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 45 + \left(30 a + 23\right)\cdot 47 + \left(3 a + 19\right)\cdot 47^{2} + \left(18 a + 26\right)\cdot 47^{3} + \left(28 a + 24\right)\cdot 47^{4} + \left(45 a + 5\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 43 a + 38 + \left(28 a + 16\right)\cdot 47 + \left(44 a + 17\right)\cdot 47^{2} + \left(32 a + 30\right)\cdot 47^{3} + \left(40 a + 20\right)\cdot 47^{4} + \left(12 a + 31\right)\cdot 47^{5} +O\left(47^{ 6 }\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 |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
