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:
| $r_{ 1 }$ |
$=$ |
$ 13 a + 18 + \left(27 a + 25\right)\cdot 31 + \left(24 a + 19\right)\cdot 31^{2} + \left(24 a + 18\right)\cdot 31^{3} + \left(18 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 23 + \left(12 a + 22\right)\cdot 31 + \left(28 a + 8\right)\cdot 31^{2} + 13 a\cdot 31^{3} + \left(2 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 25\cdot 31 + 21\cdot 31^{2} + 12\cdot 31^{3} + 17\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 a + 13 + \left(3 a + 5\right)\cdot 31 + \left(6 a + 11\right)\cdot 31^{2} + \left(6 a + 12\right)\cdot 31^{3} + \left(12 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 a + 8 + \left(18 a + 8\right)\cdot 31 + \left(2 a + 22\right)\cdot 31^{2} + \left(17 a + 30\right)\cdot 31^{3} + \left(28 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 + 5\cdot 31 + 9\cdot 31^{2} + 18\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,6,5)(2,4,3)$ |
| $(1,4)(2,5)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,6,5)(2,4,3)$ | $0$ |
| $6$ | $4$ | $(1,2,4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
