Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{3} + x + 28 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a^{2} + 5 a + 11 + \left(6 a^{2} + 30 a + 4\right)\cdot 31 + \left(16 a^{2} + 25 a + 21\right)\cdot 31^{2} + \left(10 a^{2} + 27\right)\cdot 31^{3} + \left(2 a^{2} + 28 a + 11\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 a^{2} + 28 a + 30 + \left(8 a^{2} + 5\right)\cdot 31 + \left(15 a^{2} + 7 a + 10\right)\cdot 31^{2} + \left(21 a^{2} + 14 a + 14\right)\cdot 31^{3} + \left(23 a^{2} + 10 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a^{2} + 7 a + 5 + \left(7 a^{2} + 9 a + 15\right)\cdot 31 + \left(8 a^{2} + 11 a + 5\right)\cdot 31^{2} + 16 a\cdot 31^{3} + \left(12 a^{2} + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a^{2} + 19 a + 16 + \left(17 a^{2} + 22 a + 11\right)\cdot 31 + \left(6 a^{2} + 24 a + 4\right)\cdot 31^{2} + \left(20 a^{2} + 13 a + 3\right)\cdot 31^{3} + \left(16 a^{2} + 2 a + 11\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 a + 21 + \left(3 a^{2} + 12 a + 22\right)\cdot 31 + \left(14 a^{2} + 28 a + 19\right)\cdot 31^{2} + \left(26 a^{2} + 18 a + 17\right)\cdot 31^{3} + \left(26 a^{2} + 9 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a^{2} + 13 a + 20 + \left(20 a^{2} + 9 a + 13\right)\cdot 31 + \left(11 a^{2} + 21 a + 28\right)\cdot 31^{2} + \left(23 a^{2} + 11 a + 25\right)\cdot 31^{3} + \left(10 a^{2} + 15 a + 27\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 19 a^{2} + 21 a + 13 + \left(a^{2} + 20 a + 11\right)\cdot 31 + \left(4 a^{2} + 2 a + 23\right)\cdot 31^{2} + \left(17 a^{2} + 5 a + 21\right)\cdot 31^{3} + \left(27 a^{2} + 5 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a^{2} + 22 a + 22 + \left(19 a^{2} + 15 a + 2\right)\cdot 31 + \left(19 a^{2} + 28 a + 13\right)\cdot 31^{2} + \left(6 a^{2} + 26 a + 4\right)\cdot 31^{3} + \left(26 a^{2} + 9 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 14 a^{2} + 24 a + 20 + \left(8 a^{2} + 2 a + 5\right)\cdot 31 + \left(28 a^{2} + 5 a + 29\right)\cdot 31^{2} + \left(28 a^{2} + 16 a + 8\right)\cdot 31^{3} + \left(8 a^{2} + 11 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3)(2,8)(5,7)(6,9)$ |
| $(1,5,2,4,8,7,3,9,6)$ |
| $(1,4,3)(2,7,6)(5,8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,3)(2,8)(5,7)(6,9)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,4,3)(2,7,6)(5,8,9)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,5,2,4,8,7,3,9,6)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,2,8,3,6,5,4,7,9)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,8,6,4,9,2,3,5,7)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
