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:
| $r_{ 1 }$ |
$=$ |
$ 26 a^{2} + 6 a + 5 + \left(26 a^{2} + 5 a + 10\right)\cdot 29 + \left(25 a^{2} + 12 a + 23\right)\cdot 29^{2} + \left(22 a^{2} + 10 a + 19\right)\cdot 29^{3} + \left(24 a^{2} + 5 a + 12\right)\cdot 29^{4} + \left(28 a^{2} + 20 a + 2\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a^{2} + 14 a + 16 + \left(24 a^{2} + 20 a + 7\right)\cdot 29 + \left(3 a^{2} + 13 a + 13\right)\cdot 29^{2} + \left(12 a^{2} + 12 a + 5\right)\cdot 29^{3} + \left(23 a^{2} + 18 a + 1\right)\cdot 29^{4} + \left(6 a^{2} + 27 a + 2\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 25\cdot 29 + 22\cdot 29^{2} + 5\cdot 29^{3} + 3\cdot 29^{4} + 13\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 5 a + 14 + \left(22 a^{2} + 21 a + 17\right)\cdot 29 + \left(5 a^{2} + 10 a + 1\right)\cdot 29^{2} + \left(15 a^{2} + 23 a\right)\cdot 29^{3} + \left(28 a + 20\right)\cdot 29^{4} + \left(26 a^{2} + 14 a + 27\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a^{2} + 24 a + 15 + \left(8 a^{2} + 3 a + 18\right)\cdot 29 + \left(10 a^{2} + 23 a + 7\right)\cdot 29^{2} + \left(17 a^{2} + 5 a + 22\right)\cdot 29^{3} + \left(13 a^{2} + 28 a + 27\right)\cdot 29^{4} + \left(14 a^{2} + a + 21\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a^{2} + 13 + \left(27 a^{2} + 4 a + 15\right)\cdot 29 + \left(12 a^{2} + 24 a + 1\right)\cdot 29^{2} + \left(25 a^{2} + 28 a + 4\right)\cdot 29^{3} + \left(14 a^{2} + 10\right)\cdot 29^{4} + \left(17 a^{2} + 12 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 5 a^{2} + 9 a + 6 + \left(6 a^{2} + 3 a + 21\right)\cdot 29 + \left(28 a^{2} + 3 a + 16\right)\cdot 29^{2} + \left(22 a^{2} + 6 a\right)\cdot 29^{3} + \left(9 a^{2} + 5 a + 12\right)\cdot 29^{4} + \left(22 a^{2} + 10 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,3,5,4)(6,7)$ |
| $(1,7)(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,5)(3,4)$ | $2$ |
| $56$ | $3$ | $(1,2,5)(3,4,6)$ | $0$ |
| $42$ | $4$ | $(1,3,5,4)(6,7)$ | $0$ |
| $24$ | $7$ | $(1,2,3,5,4,7,6)$ | $-1$ |
| $24$ | $7$ | $(1,5,6,3,7,2,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
