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