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