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