Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{3} + 2 x + 27 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a^{2} + 26 a + 28 + \left(10 a^{2} + 17 a + 11\right)\cdot 29 + \left(21 a^{2} + 22 a + 2\right)\cdot 29^{2} + \left(15 a^{2} + 23 a + 7\right)\cdot 29^{3} + \left(12 a^{2} + 11 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 27\cdot 29^{2} + 4\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a^{2} + 6 a + 18 + \left(20 a + 11\right)\cdot 29 + \left(17 a^{2} + 4 a + 20\right)\cdot 29^{2} + \left(3 a^{2} + 15 a + 8\right)\cdot 29^{3} + \left(3 a^{2} + 9 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a^{2} + 21 a + 2 + \left(5 a^{2} + 25 a + 5\right)\cdot 29 + \left(4 a^{2} + 4 a + 18\right)\cdot 29^{2} + \left(3 a^{2} + 14 a + 9\right)\cdot 29^{3} + \left(5 a^{2} + 14 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a^{2} + 11 a + 1 + \left(13 a^{2} + 14 a + 7\right)\cdot 29 + \left(3 a^{2} + a + 17\right)\cdot 29^{2} + \left(10 a^{2} + 20 a + 28\right)\cdot 29^{3} + \left(11 a^{2} + 2 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a^{2} + 25 a + 1 + \left(15 a^{2} + 24 a + 3\right)\cdot 29 + \left(10 a^{2} + 11 a + 2\right)\cdot 29^{2} + \left(7 a^{2} + 22 a + 4\right)\cdot 29^{3} + \left(25 a^{2} + 4 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 27 a^{2} + 27 a + 20 + \left(12 a^{2} + 12 a + 18\right)\cdot 29 + \left(a^{2} + 12 a + 28\right)\cdot 29^{2} + \left(18 a^{2} + 20 a + 27\right)\cdot 29^{3} + \left(14 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,3)(2,7)$ |
| $(2,5,6,3)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $21$ | $2$ | $(1,3)(2,7)$ | $-1$ |
| $56$ | $3$ | $(1,2,6)(3,4,5)$ | $0$ |
| $42$ | $4$ | $(2,5,6,3)(4,7)$ | $1$ |
| $24$ | $7$ | $(1,3,7,4,2,5,6)$ | $\zeta_{7}^{4} + \zeta_{7}^{2} + \zeta_{7}$ |
| $24$ | $7$ | $(1,4,6,7,5,3,2)$ | $-\zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.
