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:
| $r_{ 1 }$ |
$=$ |
$ 22 a^{2} + 2 a + 2 + \left(23 a^{2} + 25 a + 21\right)\cdot 31 + \left(24 a + 1\right)\cdot 31^{2} + \left(4 a^{2} + 21 a + 1\right)\cdot 31^{3} + \left(19 a^{2} + 27 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a^{2} + 17 a + 3 + \left(22 a^{2} + 8 a + 20\right)\cdot 31 + \left(30 a^{2} + 18 a + 21\right)\cdot 31^{2} + \left(24 a^{2} + 4 a + 4\right)\cdot 31^{3} + \left(5 a^{2} + 14 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a^{2} + 12 a + 19 + \left(16 a^{2} + 28 a + 5\right)\cdot 31 + \left(30 a^{2} + 18 a + 11\right)\cdot 31^{2} + \left(a^{2} + 4 a + 20\right)\cdot 31^{3} + \left(6 a^{2} + 20 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 8\cdot 31 + 10\cdot 31^{2} + 4\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a^{2} + 4 a + 17 + \left(18 a^{2} + 16 a + 24\right)\cdot 31 + \left(15 a^{2} + 11 a + 5\right)\cdot 31^{2} + \left(30 a^{2} + 10\right)\cdot 31^{3} + \left(5 a^{2} + 6 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a^{2} + 23 a + 5 + \left(29 a^{2} + 20 a + 1\right)\cdot 31 + \left(a^{2} + a + 7\right)\cdot 31^{2} + \left(27 a^{2} + 3 a + 18\right)\cdot 31^{3} + \left(25 a^{2} + 13 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 14 a^{2} + 4 a + 25 + \left(14 a^{2} + 25 a + 11\right)\cdot 31 + \left(13 a^{2} + 17 a + 4\right)\cdot 31^{2} + \left(4 a^{2} + 27 a + 3\right)\cdot 31^{3} + \left(30 a^{2} + 11 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2)(4,6,7,5)$ |
| $(1,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $21$ |
$2$ |
$(4,7)(5,6)$ |
$-1$ |
$-1$ |
| $56$ |
$3$ |
$(2,6,5)(3,7,4)$ |
$0$ |
$0$ |
| $42$ |
$4$ |
$(1,2)(4,6,7,5)$ |
$1$ |
$1$ |
| $24$ |
$7$ |
$(1,2,4,3,6,7,5)$ |
$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $24$ |
$7$ |
$(1,3,5,4,7,2,6)$ |
$-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
The blue line marks the conjugacy class containing complex conjugation.
