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:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 10 + \left(10 a + 22\right)\cdot 31 + \left(25 a + 18\right)\cdot 31^{2} + \left(2 a + 26\right)\cdot 31^{3} + \left(14 a + 2\right)\cdot 31^{4} + \left(11 a + 17\right)\cdot 31^{5} + \left(29 a + 29\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 25\cdot 31 + 15\cdot 31^{2} + 28\cdot 31^{3} + 30\cdot 31^{4} + 18\cdot 31^{5} + 17\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a + 22 + \left(7 a + 21\right)\cdot 31 + \left(26 a + 25\right)\cdot 31^{2} + \left(29 a + 19\right)\cdot 31^{3} + \left(24 a + 6\right)\cdot 31^{4} + \left(15 a + 10\right)\cdot 31^{5} + \left(6 a + 26\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 8 + \left(20 a + 14\right)\cdot 31 + \left(5 a + 27\right)\cdot 31^{2} + \left(28 a + 6\right)\cdot 31^{3} + \left(16 a + 28\right)\cdot 31^{4} + \left(19 a + 25\right)\cdot 31^{5} + \left(a + 14\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 19 + \left(23 a + 22\right)\cdot 31 + \left(4 a + 8\right)\cdot 31^{2} + \left(a + 22\right)\cdot 31^{3} + \left(6 a + 26\right)\cdot 31^{4} + \left(15 a + 16\right)\cdot 31^{5} + \left(24 a + 23\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 17\cdot 31 + 27\cdot 31^{2} + 19\cdot 31^{3} + 28\cdot 31^{4} + 3\cdot 31^{5} + 12\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)(3,5,6)$ |
| $(1,3)(2,6)(4,5)$ |
| $(2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(2,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,2,4)(3,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,3,2,5,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
