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