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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 14 + \left(29 a + 15\right)\cdot 31 + \left(17 a + 17\right)\cdot 31^{2} + \left(7 a + 9\right)\cdot 31^{3} + \left(20 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 a + 21 + \left(3 a + 9\right)\cdot 31 + \left(13 a + 15\right)\cdot 31^{2} + \left(27 a + 23\right)\cdot 31^{3} + 9 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 24\cdot 31 + 27\cdot 31^{2} + 2\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 27 a + 29 + \left(27 a + 11\right)\cdot 31 + \left(17 a + 7\right)\cdot 31^{2} + \left(3 a + 3\right)\cdot 31^{3} + \left(21 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 22\cdot 31 + 16\cdot 31^{3} + 29\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 18 + \left(a + 9\right)\cdot 31 + \left(13 a + 24\right)\cdot 31^{2} + \left(23 a + 6\right)\cdot 31^{3} + \left(10 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,2,6,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $15$ | $2$ | $(1,3)(4,6)$ | $-2$ |
| $20$ | $3$ | $(1,4,6)(2,5,3)$ | $0$ |
| $30$ | $4$ | $(3,4,6,5)$ | $0$ |
| $24$ | $5$ | $(1,3,5,6,2)$ | $1$ |
| $20$ | $6$ | $(1,5,4,3,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
