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:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 46 + \left(15 a + 44\right)\cdot 47 + \left(26 a + 17\right)\cdot 47^{2} + \left(45 a + 45\right)\cdot 47^{3} + \left(39 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 19 + \left(43 a + 7\right)\cdot 47 + \left(31 a + 22\right)\cdot 47^{2} + \left(21 a + 12\right)\cdot 47^{3} + \left(46 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 43 + \left(3 a + 34\right)\cdot 47 + \left(15 a + 42\right)\cdot 47^{2} + \left(25 a + 23\right)\cdot 47^{3} + 18\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 14\cdot 47 + 10\cdot 47^{2} + 17\cdot 47^{3} + 5\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 4\cdot 47 + 40\cdot 47^{2} + 25\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 38 + \left(31 a + 33\right)\cdot 47 + \left(20 a + 7\right)\cdot 47^{2} + \left(a + 16\right)\cdot 47^{3} + \left(7 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
