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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 19 + \left(27 a + 24\right)\cdot 31 + \left(23 a + 14\right)\cdot 31^{2} + \left(8 a + 19\right)\cdot 31^{3} + \left(27 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 21 + \left(27 a + 23\right)\cdot 31 + \left(23 a + 20\right)\cdot 31^{2} + \left(8 a + 8\right)\cdot 31^{3} + \left(27 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 1 + \left(3 a + 27\right)\cdot 31 + \left(7 a + 3\right)\cdot 31^{2} + \left(22 a + 13\right)\cdot 31^{3} + \left(3 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 16 + \left(3 a + 12\right)\cdot 31 + \left(7 a + 16\right)\cdot 31^{2} + \left(22 a + 21\right)\cdot 31^{3} + \left(3 a + 9\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 3 + \left(3 a + 26\right)\cdot 31 + \left(7 a + 9\right)\cdot 31^{2} + \left(22 a + 2\right)\cdot 31^{3} + \left(3 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 3 + \left(27 a + 10\right)\cdot 31 + \left(23 a + 27\right)\cdot 31^{2} + \left(8 a + 27\right)\cdot 31^{3} + \left(27 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,2)(3,4,5)$ |
| $(1,3)(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,3,6,5)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,5,6,3,2,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
