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