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