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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 7 + \left(26 a + 16\right)\cdot 31 + \left(7 a + 5\right)\cdot 31^{2} + 19 a\cdot 31^{3} + \left(14 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 25\cdot 31 + 21\cdot 31^{2} + 14\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 29 + \left(5 a + 26\right)\cdot 31 + \left(22 a + 26\right)\cdot 31^{2} + 25 a\cdot 31^{3} + \left(2 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 24 + \left(4 a + 14\right)\cdot 31 + \left(23 a + 25\right)\cdot 31^{2} + \left(11 a + 30\right)\cdot 31^{3} + \left(16 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 5\cdot 31 + 9\cdot 31^{2} + 16\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 2 + \left(25 a + 4\right)\cdot 31 + \left(8 a + 4\right)\cdot 31^{2} + \left(5 a + 30\right)\cdot 31^{3} + \left(28 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,2,4,5)$ |
| $(1,5,3)(2,6,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,2,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
