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^{3} + x + 28 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 a^{2} + 30 a + 18 + \left(28 a^{2} + 22 a + 11\right)\cdot 31 + \left(28 a^{2} + 2 a + 10\right)\cdot 31^{2} + \left(9 a^{2} + a + 25\right)\cdot 31^{3} + \left(3 a^{2} + 21 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 14 + \left(28 a^{2} + 28 a + 21\right)\cdot 31 + \left(28 a^{2} + 29 a + 6\right)\cdot 31^{2} + \left(23 a^{2} + 17 a + 14\right)\cdot 31^{3} + \left(25 a^{2} + 13 a + 1\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 11 + \left(25 a^{2} + 18 a + 19\right)\cdot 31 + \left(18 a^{2} + 18 a + 3\right)\cdot 31^{2} + \left(13 a^{2} + 9 a + 7\right)\cdot 31^{3} + \left(26 a^{2} + 22 a + 1\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a^{2} + 7 a + 15 + \left(23 a^{2} + 12 a + 18\right)\cdot 31 + \left(30 a^{2} + 27 a + 28\right)\cdot 31^{2} + \left(12 a^{2} + 25 a + 6\right)\cdot 31^{3} + \left(27 a^{2} + 3 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a^{2} + 13 a + 4 + \left(8 a^{2} + 20 a + 8\right)\cdot 31 + \left(14 a^{2} + 9 a + 21\right)\cdot 31^{2} + \left(7 a^{2} + 20 a + 23\right)\cdot 31^{3} + \left(a^{2} + 18 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a^{2} + 9 a + 13 + \left(10 a^{2} + 21 a + 30\right)\cdot 31 + \left(2 a^{2} + 4 a + 19\right)\cdot 31^{2} + \left(25 a^{2} + 18 a + 4\right)\cdot 31^{3} + \left(8 a^{2} + 13 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 + 14\cdot 31 + 2\cdot 31^{2} + 11\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,6)(3,4)$ |
| $(1,3,7,2)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $7$ |
| $21$ | $2$ | $(1,7)(2,3)$ | $-1$ |
| $56$ | $3$ | $(1,4,7)(2,5,3)$ | $1$ |
| $42$ | $4$ | $(1,3,7,2)(5,6)$ | $-1$ |
| $24$ | $7$ | $(1,4,3,7,2,6,5)$ | $0$ |
| $24$ | $7$ | $(1,7,5,3,6,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
