Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ a^{2} + 3 a + 23 + \left(15 a^{2} + 9 a + 15\right)\cdot 29 + \left(6 a^{2} + 8 a + 7\right)\cdot 29^{2} + \left(25 a^{2} + 27 a + 8\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 20\right)\cdot 29^{4} + \left(a^{2} + 13 a + 10\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a^{2} + 28 a + 8 + \left(25 a^{2} + 8 a + 15\right)\cdot 29 + \left(4 a^{2} + 23 a + 21\right)\cdot 29^{2} + \left(18 a^{2} + 28 a + 15\right)\cdot 29^{3} + \left(26 a^{2} + 19 a\right)\cdot 29^{4} + \left(15 a^{2} + 2 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a^{2} + 28 a + 19 + \left(21 a^{2} + 22 a + 19\right)\cdot 29 + \left(2 a^{2} + 2 a + 18\right)\cdot 29^{2} + \left(a^{2} + 25 a + 2\right)\cdot 29^{3} + \left(5 a^{2} + 18 a + 20\right)\cdot 29^{4} + \left(2 a^{2} + 9 a + 28\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 12 + \left(17 a^{2} + 25 a + 18\right)\cdot 29 + \left(21 a^{2} + 4 a + 27\right)\cdot 29^{2} + \left(17 a^{2} + 5 a + 7\right)\cdot 29^{3} + \left(12 a^{2} + 17 a\right)\cdot 29^{4} + \left(13 a^{2} + 9 a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 10\cdot 29 + 16\cdot 29^{2} + 14\cdot 29^{3} + 9\cdot 29^{4} + 11\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a^{2} + 2 a + 19 + \left(11 a^{2} + 26 a + 25\right)\cdot 29 + \left(21 a^{2} + 2 a + 4\right)\cdot 29^{2} + \left(9 a^{2} + 4 a + 14\right)\cdot 29^{3} + \left(26 a^{2} + 19 a + 19\right)\cdot 29^{4} + \left(10 a^{2} + 16 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 a^{2} + 1 + \left(25 a^{2} + 23 a + 11\right)\cdot 29 + \left(15 a + 19\right)\cdot 29^{2} + \left(15 a^{2} + 25 a + 23\right)\cdot 29^{3} + \left(10 a^{2} + 17 a + 16\right)\cdot 29^{4} + \left(14 a^{2} + 5 a + 18\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,3,7,5)(2,6)$ |
| $(1,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,4)(3,6)$ | $2$ |
| $56$ | $3$ | $(1,2,7)(3,5,4)$ | $0$ |
| $42$ | $4$ | $(1,3,7,5)(2,6)$ | $0$ |
| $24$ | $7$ | $(1,4,3,2,6,7,5)$ | $-1$ |
| $24$ | $7$ | $(1,2,5,3,7,4,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
