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:
| $r_{ 1 }$ |
$=$ |
$ a + 3 + \left(2 a + 8\right)\cdot 13 + \left(6 a + 6\right)\cdot 13^{2} + \left(12 a + 9\right)\cdot 13^{3} + \left(8 a + 8\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 7 + \left(2 a + 1\right)\cdot 13 + \left(4 a + 8\right)\cdot 13^{2} + \left(11 a + 3\right)\cdot 13^{3} + \left(a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 10 + 10 a\cdot 13 + \left(8 a + 10\right)\cdot 13^{2} + \left(a + 10\right)\cdot 13^{3} + \left(11 a + 8\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 a + 4 + \left(10 a + 9\right)\cdot 13 + \left(6 a + 10\right)\cdot 13^{2} + 2\cdot 13^{3} + \left(4 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 10\cdot 13 + 7\cdot 13^{2} + 11\cdot 13^{3} + 11\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 8\cdot 13 + 8\cdot 13^{2} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(1,4)$ |
$-2$ |
| $9$ |
$2$ |
$(1,4)(2,3)$ |
$0$ |
| $4$ |
$3$ |
$(1,4,6)(2,3,5)$ |
$-2$ |
| $4$ |
$3$ |
$(2,3,5)$ |
$1$ |
| $18$ |
$4$ |
$(1,3,4,2)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,2,4,3,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,4)(2,3,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
