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:
| $r_{ 1 }$ |
$=$ |
$ 17 a + 15 + \left(41 a + 5\right)\cdot 47 + \left(16 a + 23\right)\cdot 47^{2} + \left(22 a + 6\right)\cdot 47^{3} + \left(41 a + 46\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 a + 22 + \left(12 a + 26\right)\cdot 47 + \left(18 a + 26\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(38 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 2 + \left(5 a + 24\right)\cdot 47 + \left(30 a + 15\right)\cdot 47^{2} + \left(24 a + 34\right)\cdot 47^{3} + \left(5 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 20 + \left(34 a + 6\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(17 a + 38\right)\cdot 47^{3} + \left(8 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 22 + \left(36 a + 26\right)\cdot 47 + \left(23 a + 15\right)\cdot 47^{2} + \left(4 a + 10\right)\cdot 47^{3} + \left(6 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 2\cdot 47 + 30\cdot 47^{2} + 11\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 a + 26 + \left(10 a + 2\right)\cdot 47 + \left(23 a + 27\right)\cdot 47^{2} + \left(42 a + 42\right)\cdot 47^{3} + \left(40 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,7)(2,3)(4,6)$ |
| $(1,3)(2,4)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $7$ | $2$ | $(1,7)(2,3)(4,6)$ | $0$ |
| $2$ | $7$ | $(1,5,7,3,4,6,2)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
| $2$ | $7$ | $(1,7,4,2,5,3,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
| $2$ | $7$ | $(1,3,2,7,6,5,4)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
