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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 3 + \left(4 a + 27\right)\cdot 31 + \left(11 a + 28\right)\cdot 31^{2} + \left(19 a + 28\right)\cdot 31^{3} + \left(11 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 15 + \left(18 a + 24\right)\cdot 31 + \left(12 a + 2\right)\cdot 31^{2} + \left(6 a + 1\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 13 + \left(26 a + 30\right)\cdot 31 + \left(19 a + 15\right)\cdot 31^{2} + \left(11 a + 25\right)\cdot 31^{3} + \left(19 a + 12\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 19\cdot 31 + 4\cdot 31^{2} + 5\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 29 + \left(12 a + 22\right)\cdot 31 + \left(18 a + 9\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(2 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
