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