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