Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 a + 28 + \left(25 a + 20\right)\cdot 31 + \left(20 a + 15\right)\cdot 31^{2} + \left(16 a + 19\right)\cdot 31^{3} + \left(20 a + 14\right)\cdot 31^{4} + \left(22 a + 25\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 10 + \left(18 a + 26\right)\cdot 31 + \left(8 a + 12\right)\cdot 31^{2} + \left(11 a + 21\right)\cdot 31^{3} + 4\cdot 31^{4} + \left(16 a + 22\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 a + 13 + \left(5 a + 1\right)\cdot 31 + \left(10 a + 1\right)\cdot 31^{2} + \left(14 a + 1\right)\cdot 31^{3} + \left(10 a + 8\right)\cdot 31^{4} + \left(8 a + 19\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 8\cdot 31 + 14\cdot 31^{2} + 10\cdot 31^{3} + 8\cdot 31^{4} + 17\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 30 + \left(12 a + 21\right)\cdot 31 + \left(22 a + 11\right)\cdot 31^{2} + \left(19 a + 4\right)\cdot 31^{3} + \left(30 a + 25\right)\cdot 31^{4} + \left(14 a + 22\right)\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 13\cdot 31 + 6\cdot 31^{2} + 5\cdot 31^{3} + 31^{4} + 17\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(3,4)$ |
| $(1,2,4,5,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-2$ |
| $3$ | $2$ | $(2,6)(3,4)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,2,4,5,3,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
