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