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:
| $r_{ 1 }$ |
$=$ |
$ 26 + 7\cdot 31 + 27\cdot 31^{2} + 24\cdot 31^{4} + 22\cdot 31^{5} + 9\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 6 + \left(8 a + 29\right)\cdot 31 + 30\cdot 31^{2} + \left(19 a + 8\right)\cdot 31^{3} + \left(27 a + 18\right)\cdot 31^{4} + \left(11 a + 1\right)\cdot 31^{5} + \left(28 a + 27\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 26 + \left(22 a + 4\right)\cdot 31 + \left(30 a + 23\right)\cdot 31^{2} + \left(11 a + 15\right)\cdot 31^{3} + \left(3 a + 23\right)\cdot 31^{4} + \left(19 a + 28\right)\cdot 31^{5} + \left(2 a + 9\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 25 + \left(23 a + 1\right)\cdot 31 + \left(24 a + 4\right)\cdot 31^{2} + \left(2 a + 8\right)\cdot 31^{3} + \left(4 a + 10\right)\cdot 31^{4} + \left(10 a + 4\right)\cdot 31^{5} + \left(a + 8\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 4 + \left(21 a + 26\right)\cdot 31 + \left(9 a + 30\right)\cdot 31^{2} + \left(7 a + 12\right)\cdot 31^{3} + \left(30 a + 7\right)\cdot 31^{4} + \left(7 a + 16\right)\cdot 31^{5} + \left(6 a + 16\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 + 5\cdot 31 + 10\cdot 31^{2} + 8\cdot 31^{3} + 26\cdot 31^{4} + 27\cdot 31^{5} + 30\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 a + 10 + \left(9 a + 3\right)\cdot 31 + \left(21 a + 29\right)\cdot 31^{2} + \left(23 a + 17\right)\cdot 31^{3} + 29\cdot 31^{4} + \left(23 a + 1\right)\cdot 31^{5} + \left(24 a + 21\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 29 a + 29 + \left(7 a + 14\right)\cdot 31 + \left(6 a + 30\right)\cdot 31^{2} + \left(28 a + 19\right)\cdot 31^{3} + \left(26 a + 15\right)\cdot 31^{4} + \left(20 a + 20\right)\cdot 31^{5} + 29 a\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6,5)(2,8,4,3)$ |
| $(1,3,6,8)(2,7,4,5)$ |
| $(1,6)(2,4)(3,8)(5,7)$ |
| $(1,3,5)(6,8,7)$ |
| $(1,7)(3,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-4$ |
| $12$ | $2$ | $(1,7)(3,8)(5,6)$ | $0$ |
| $8$ | $3$ | $(1,8,4)(2,6,3)$ | $1$ |
| $6$ | $4$ | $(1,7,6,5)(2,8,4,3)$ | $0$ |
| $8$ | $6$ | $(1,2,8,6,4,3)(5,7)$ | $-1$ |
| $6$ | $8$ | $(1,4,5,8,6,2,7,3)$ | $0$ |
| $6$ | $8$ | $(1,2,5,3,6,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
