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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 11 + \left(17 a + 21\right)\cdot 31 + \left(5 a + 1\right)\cdot 31^{2} + \left(22 a + 16\right)\cdot 31^{3} + 5\cdot 31^{4} + \left(13 a + 26\right)\cdot 31^{5} + 8\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 a + 10 + \left(13 a + 10\right)\cdot 31 + \left(25 a + 26\right)\cdot 31^{2} + \left(8 a + 23\right)\cdot 31^{3} + \left(30 a + 15\right)\cdot 31^{4} + \left(17 a + 20\right)\cdot 31^{5} + \left(30 a + 27\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 9 + \left(10 a + 13\right)\cdot 31 + \left(11 a + 15\right)\cdot 31^{2} + \left(18 a + 29\right)\cdot 31^{3} + \left(a + 7\right)\cdot 31^{4} + \left(27 a + 22\right)\cdot 31^{5} + \left(29 a + 5\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 10 + \left(30 a + 24\right)\cdot 31 + \left(23 a + 17\right)\cdot 31^{2} + 26\cdot 31^{3} + \left(6 a + 29\right)\cdot 31^{4} + \left(29 a + 25\right)\cdot 31^{5} + \left(30 a + 20\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 19 + \left(20 a + 28\right)\cdot 31 + \left(19 a + 27\right)\cdot 31^{2} + \left(12 a + 23\right)\cdot 31^{3} + \left(29 a + 23\right)\cdot 31^{4} + \left(3 a + 12\right)\cdot 31^{5} + \left(a + 7\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 2 a + 6 + 26\cdot 31 + \left(7 a + 3\right)\cdot 31^{2} + \left(30 a + 4\right)\cdot 31^{3} + \left(24 a + 10\right)\cdot 31^{4} + \left(a + 16\right)\cdot 31^{5} + 22\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5,4)$ |
| $(1,3,6)$ |
| $(1,5,6,2,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,3)(2,5,4)$ | $-1$ |
| $3$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
| $3$ | $6$ | $(1,4,3,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
