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