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