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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 22 + \left(17 a + 28\right)\cdot 31 + \left(24 a + 7\right)\cdot 31^{2} + \left(29 a + 12\right)\cdot 31^{3} + \left(27 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 19 + \left(20 a + 12\right)\cdot 31 + \left(2 a + 25\right)\cdot 31^{2} + \left(20 a + 19\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 21 + \left(13 a + 17\right)\cdot 31 + \left(6 a + 8\right)\cdot 31^{2} + \left(a + 16\right)\cdot 31^{3} + 3 a\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a + 11 + \left(10 a + 27\right)\cdot 31 + \left(28 a + 9\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + \left(21 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 6\cdot 31 + 10\cdot 31^{2} + 18\cdot 31^{3} + 5\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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
