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