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:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 23 + \left(17 a + 17\right)\cdot 31 + \left(30 a + 24\right)\cdot 31^{2} + \left(28 a + 1\right)\cdot 31^{3} + \left(17 a + 12\right)\cdot 31^{4} + \left(3 a + 5\right)\cdot 31^{5} + \left(21 a + 27\right)\cdot 31^{6} + \left(26 a + 14\right)\cdot 31^{7} + \left(25 a + 18\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 8 + \left(26 a + 18\right)\cdot 31 + \left(16 a + 11\right)\cdot 31^{2} + \left(27 a + 9\right)\cdot 31^{3} + \left(14 a + 14\right)\cdot 31^{4} + \left(6 a + 14\right)\cdot 31^{5} + \left(6 a + 13\right)\cdot 31^{6} + \left(24 a + 23\right)\cdot 31^{7} + \left(5 a + 1\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 12\cdot 31 + 23\cdot 31^{2} + 4\cdot 31^{3} + 15\cdot 31^{4} + 17\cdot 31^{5} + 31^{6} + 13\cdot 31^{7} + 5\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 a + 6 + \left(26 a + 14\right)\cdot 31 + \left(16 a + 12\right)\cdot 31^{2} + \left(27 a + 14\right)\cdot 31^{3} + \left(14 a + 14\right)\cdot 31^{4} + \left(6 a + 18\right)\cdot 31^{5} + \left(6 a + 11\right)\cdot 31^{6} + \left(24 a + 27\right)\cdot 31^{7} + \left(5 a + 10\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 8 + \left(13 a + 13\right)\cdot 31 + 6\cdot 31^{2} + \left(2 a + 29\right)\cdot 31^{3} + \left(13 a + 18\right)\cdot 31^{4} + \left(27 a + 25\right)\cdot 31^{5} + \left(9 a + 3\right)\cdot 31^{6} + \left(4 a + 16\right)\cdot 31^{7} + \left(5 a + 12\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 23 + \left(4 a + 12\right)\cdot 31 + \left(14 a + 19\right)\cdot 31^{2} + \left(3 a + 21\right)\cdot 31^{3} + \left(16 a + 16\right)\cdot 31^{4} + \left(24 a + 16\right)\cdot 31^{5} + \left(24 a + 17\right)\cdot 31^{6} + \left(6 a + 7\right)\cdot 31^{7} + \left(25 a + 29\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 12 + 18\cdot 31 + 7\cdot 31^{2} + 26\cdot 31^{3} + 15\cdot 31^{4} + 13\cdot 31^{5} + 29\cdot 31^{6} + 17\cdot 31^{7} + 25\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 7 a + 25 + \left(4 a + 16\right)\cdot 31 + \left(14 a + 18\right)\cdot 31^{2} + \left(3 a + 16\right)\cdot 31^{3} + \left(16 a + 16\right)\cdot 31^{4} + \left(24 a + 12\right)\cdot 31^{5} + \left(24 a + 19\right)\cdot 31^{6} + \left(6 a + 3\right)\cdot 31^{7} + \left(25 a + 20\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,5,2)(3,8,7,4)$ |
| $(1,3,5,7)(2,8,6,4)$ |
| $(1,2,4)(5,6,8)$ |
| $(1,5)(2,8)(4,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,5)(2,8)(4,6)$ | $0$ |
| $8$ | $3$ | $(1,2,4)(5,6,8)$ | $-1$ |
| $6$ | $4$ | $(1,6,5,2)(3,8,7,4)$ | $0$ |
| $8$ | $6$ | $(1,6,4,5,2,8)(3,7)$ | $1$ |
| $6$ | $8$ | $(1,4,3,2,5,8,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,8,3,6,5,4,7,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
