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