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:
| $r_{ 1 }$ |
$=$ |
$ 43 + 26\cdot 47 + 25\cdot 47^{2} + 22\cdot 47^{3} + 19\cdot 47^{4} + 44\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 38 + \left(38 a + 8\right)\cdot 47 + \left(13 a + 19\right)\cdot 47^{2} + \left(23 a + 39\right)\cdot 47^{3} + 14 a\cdot 47^{4} + \left(45 a + 35\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 18 + \left(8 a + 2\right)\cdot 47 + \left(33 a + 8\right)\cdot 47^{2} + \left(23 a + 25\right)\cdot 47^{3} + \left(32 a + 6\right)\cdot 47^{4} + \left(a + 17\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 42 + \left(25 a + 45\right)\cdot 47 + \left(16 a + 15\right)\cdot 47^{2} + \left(9 a + 38\right)\cdot 47^{3} + \left(38 a + 42\right)\cdot 47^{4} + \left(24 a + 42\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 19 + \left(21 a + 37\right)\cdot 47 + \left(30 a + 23\right)\cdot 47^{2} + \left(37 a + 40\right)\cdot 47^{3} + \left(8 a + 15\right)\cdot 47^{4} + \left(22 a + 7\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 19\cdot 47 + 47^{2} + 22\cdot 47^{3} + 8\cdot 47^{4} + 41\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)$ |
| $(4,5)$ |
| $(2,3)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,3)$ | $-1$ |
| $4$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $4$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $4$ | $6$ | $(1,2,4,6,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
