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