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