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