Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{5} + 3 x + 27 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 a^{4} + 15 a^{3} + 24 a^{2} + 28 a + 2 + \left(22 a^{4} + 13 a^{2} + 15\right)\cdot 29 + \left(23 a^{4} + 26 a^{3} + 24 a^{2} + 20 a + 18\right)\cdot 29^{2} + \left(25 a^{4} + 7 a^{3} + 23 a^{2} + 20 a + 28\right)\cdot 29^{3} + \left(2 a^{4} + 28 a^{3} + 16 a^{2} + 23 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a^{4} + 15 a^{3} + 24 a^{2} + 28 a + 5 + \left(22 a^{4} + 13 a^{2} + 10\right)\cdot 29 + \left(23 a^{4} + 26 a^{3} + 24 a^{2} + 20 a + 14\right)\cdot 29^{2} + \left(25 a^{4} + 7 a^{3} + 23 a^{2} + 20 a + 2\right)\cdot 29^{3} + \left(2 a^{4} + 28 a^{3} + 16 a^{2} + 23 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a^{4} + 24 a^{3} + 27 a^{2} + 18 a + 11 + \left(28 a^{4} + 19 a^{3} + 19 a^{2} + 3 a + 1\right)\cdot 29 + \left(24 a^{3} + a^{2} + 15 a + 16\right)\cdot 29^{2} + \left(27 a^{4} + 3 a^{3} + 12 a^{2} + 16 a + 2\right)\cdot 29^{3} + \left(22 a^{4} + 17 a^{3} + 5 a^{2} + 27 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a^{4} + 24 a^{3} + 27 a^{2} + 18 a + 14 + \left(28 a^{4} + 19 a^{3} + 19 a^{2} + 3 a + 25\right)\cdot 29 + \left(24 a^{3} + a^{2} + 15 a + 11\right)\cdot 29^{2} + \left(27 a^{4} + 3 a^{3} + 12 a^{2} + 16 a + 5\right)\cdot 29^{3} + \left(22 a^{4} + 17 a^{3} + 5 a^{2} + 27 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a^{4} + 11 a^{3} + 23 a^{2} + 9 a + 9 + \left(4 a^{4} + 23 a^{3} + 23 a^{2} + 18 a + 25\right)\cdot 29 + \left(13 a^{4} + 5 a^{3} + 12 a^{2} + a + 27\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 5 a^{2} + 12\right)\cdot 29^{3} + \left(7 a^{4} + 26 a^{3} + 8 a^{2} + 22 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a^{4} + 11 a^{3} + 23 a^{2} + 9 a + 12 + \left(4 a^{4} + 23 a^{3} + 23 a^{2} + 18 a + 20\right)\cdot 29 + \left(13 a^{4} + 5 a^{3} + 12 a^{2} + a + 23\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 5 a^{2} + 15\right)\cdot 29^{3} + \left(7 a^{4} + 26 a^{3} + 8 a^{2} + 22 a + 4\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 a^{4} + 17 a^{3} + 16 a^{2} + 24 a + 8 + \left(23 a^{4} + 4 a^{3} + 9 a^{2} + 6 a + 7\right)\cdot 29 + \left(24 a^{4} + 15 a^{2} + 23 a + 15\right)\cdot 29^{2} + \left(16 a^{4} + 28 a^{3} + 28 a^{2} + 24\right)\cdot 29^{3} + \left(4 a^{4} + 15 a^{3} + 26 a^{2} + 28 a + 12\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 22 a^{4} + 17 a^{3} + 16 a^{2} + 24 a + 11 + \left(23 a^{4} + 4 a^{3} + 9 a^{2} + 6 a + 2\right)\cdot 29 + \left(24 a^{4} + 15 a^{2} + 23 a + 11\right)\cdot 29^{2} + \left(16 a^{4} + 28 a^{3} + 28 a^{2} + 27\right)\cdot 29^{3} + \left(4 a^{4} + 15 a^{3} + 26 a^{2} + 28 a + 14\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 24 a^{4} + 20 a^{3} + 26 a^{2} + 8 a + 7 + \left(7 a^{4} + 9 a^{3} + 19 a^{2} + 28 a + 21\right)\cdot 29 + \left(24 a^{4} + a^{3} + 3 a^{2} + 26 a + 19\right)\cdot 29^{2} + \left(2 a^{4} + 4 a^{3} + 17 a^{2} + 19 a + 25\right)\cdot 29^{3} + \left(20 a^{4} + 28 a^{3} + 14 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 10 }$ |
$=$ |
$ 24 a^{4} + 20 a^{3} + 26 a^{2} + 8 a + 10 + \left(7 a^{4} + 9 a^{3} + 19 a^{2} + 28 a + 16\right)\cdot 29 + \left(24 a^{4} + a^{3} + 3 a^{2} + 26 a + 15\right)\cdot 29^{2} + \left(2 a^{4} + 4 a^{3} + 17 a^{2} + 19 a + 28\right)\cdot 29^{3} + \left(20 a^{4} + 28 a^{3} + 14 a + 28\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 10 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)(9,10)$ |
| $(1,3,7,5,9)(2,4,8,6,10)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 10 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)(9,10)$ |
$-1$ |
$-1$ |
$-1$ |
$-1$ |
| $1$ |
$5$ |
$(1,3,7,5,9)(2,4,8,6,10)$ |
$\zeta_{5}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}^{3}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ |
$5$ |
$(1,7,9,3,5)(2,8,10,4,6)$ |
$\zeta_{5}^{2}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}$ |
$\zeta_{5}^{3}$ |
| $1$ |
$5$ |
$(1,5,3,9,7)(2,6,4,10,8)$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{2}$ |
| $1$ |
$5$ |
$(1,9,5,7,3)(2,10,6,8,4)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}$ |
| $1$ |
$10$ |
$(1,4,7,6,9,2,3,8,5,10)$ |
$-\zeta_{5}$ |
$-\zeta_{5}^{2}$ |
$-\zeta_{5}^{3}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
| $1$ |
$10$ |
$(1,6,3,10,7,2,5,4,9,8)$ |
$-\zeta_{5}^{3}$ |
$-\zeta_{5}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$-\zeta_{5}^{2}$ |
| $1$ |
$10$ |
$(1,8,9,4,5,2,7,10,3,6)$ |
$-\zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$-\zeta_{5}$ |
$-\zeta_{5}^{3}$ |
| $1$ |
$10$ |
$(1,10,5,8,3,2,9,6,7,4)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$-\zeta_{5}^{3}$ |
$-\zeta_{5}^{2}$ |
$-\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.
