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:
| $r_{ 1 }$ |
$=$ |
$ a + 38 + \left(25 a + 36\right)\cdot 47 + \left(31 a + 27\right)\cdot 47^{2} + \left(13 a + 7\right)\cdot 47^{3} + \left(13 a + 2\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 2 + \left(16 a + 37\right)\cdot 47 + 21\cdot 47^{2} + \left(18 a + 11\right)\cdot 47^{3} + \left(35 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 16 + \left(45 a + 35\right)\cdot 47 + \left(43 a + 11\right)\cdot 47^{2} + \left(23 a + 10\right)\cdot 47^{3} + \left(2 a + 24\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 34 + \left(a + 22\right)\cdot 47 + \left(3 a + 7\right)\cdot 47^{2} + \left(23 a + 14\right)\cdot 47^{3} + \left(44 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 a + 40 + \left(21 a + 38\right)\cdot 47 + \left(15 a + 18\right)\cdot 47^{2} + \left(33 a + 3\right)\cdot 47^{3} + \left(33 a + 15\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 a + 12 + \left(30 a + 17\right)\cdot 47 + \left(46 a + 6\right)\cdot 47^{2} + 28 a\cdot 47^{3} + \left(11 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,4)(3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,2,4)(3,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,2)(3,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,4,5,2,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,5,4,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
