Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 18 + \left(9 a + 15\right)\cdot 29 + \left(27 a + 23\right)\cdot 29^{2} + \left(a + 8\right)\cdot 29^{3} + \left(10 a + 19\right)\cdot 29^{4} + \left(19 a + 14\right)\cdot 29^{5} + \left(21 a + 13\right)\cdot 29^{6} + \left(18 a + 7\right)\cdot 29^{7} + \left(11 a + 9\right)\cdot 29^{8} + \left(20 a + 27\right)\cdot 29^{9} + \left(8 a + 2\right)\cdot 29^{10} + 18\cdot 29^{11} + 14\cdot 29^{12} + \left(12 a + 13\right)\cdot 29^{13} +O\left(29^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 26\cdot 29 + 2\cdot 29^{2} + 19\cdot 29^{3} + 10\cdot 29^{4} + 15\cdot 29^{6} + 20\cdot 29^{7} + 25\cdot 29^{8} + 9\cdot 29^{9} + 25\cdot 29^{10} + 3\cdot 29^{11} + 27\cdot 29^{12} + 16\cdot 29^{13} +O\left(29^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 12 + \left(19 a + 13\right)\cdot 29 + \left(a + 5\right)\cdot 29^{2} + \left(27 a + 20\right)\cdot 29^{3} + \left(18 a + 9\right)\cdot 29^{4} + \left(9 a + 14\right)\cdot 29^{5} + \left(7 a + 15\right)\cdot 29^{6} + \left(10 a + 21\right)\cdot 29^{7} + \left(17 a + 19\right)\cdot 29^{8} + \left(8 a + 1\right)\cdot 29^{9} + \left(20 a + 26\right)\cdot 29^{10} + \left(28 a + 10\right)\cdot 29^{11} + \left(28 a + 14\right)\cdot 29^{12} + \left(16 a + 15\right)\cdot 29^{13} +O\left(29^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 5 + \left(15 a + 22\right)\cdot 29 + \left(7 a + 17\right)\cdot 29^{2} + \left(9 a + 9\right)\cdot 29^{3} + \left(5 a + 20\right)\cdot 29^{4} + \left(5 a + 18\right)\cdot 29^{5} + \left(7 a + 13\right)\cdot 29^{6} + \left(7 a + 14\right)\cdot 29^{7} + \left(13 a + 28\right)\cdot 29^{8} + \left(26 a + 12\right)\cdot 29^{9} + \left(19 a + 21\right)\cdot 29^{10} + \left(25 a + 3\right)\cdot 29^{11} + \left(28 a + 13\right)\cdot 29^{12} + \left(16 a + 15\right)\cdot 29^{13} +O\left(29^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 2\cdot 29 + 26\cdot 29^{2} + 9\cdot 29^{3} + 18\cdot 29^{4} + 28\cdot 29^{5} + 13\cdot 29^{6} + 8\cdot 29^{7} + 3\cdot 29^{8} + 19\cdot 29^{9} + 3\cdot 29^{10} + 25\cdot 29^{11} + 29^{12} + 12\cdot 29^{13} +O\left(29^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 25 + \left(13 a + 6\right)\cdot 29 + \left(21 a + 11\right)\cdot 29^{2} + \left(19 a + 19\right)\cdot 29^{3} + \left(23 a + 8\right)\cdot 29^{4} + \left(23 a + 10\right)\cdot 29^{5} + \left(21 a + 15\right)\cdot 29^{6} + \left(21 a + 14\right)\cdot 29^{7} + 15 a\cdot 29^{8} + \left(2 a + 16\right)\cdot 29^{9} + \left(9 a + 7\right)\cdot 29^{10} + \left(3 a + 25\right)\cdot 29^{11} + 15\cdot 29^{12} + \left(12 a + 13\right)\cdot 29^{13} +O\left(29^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)$ |
| $(1,2)(3,5)$ |
| $(1,2,4)(3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(2,4)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,3,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,3)(2,6,5,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,5,6,3,2,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
