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