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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 9 + \left(10 a + 8\right)\cdot 13 + \left(9 a + 7\right)\cdot 13^{2} + \left(10 a + 5\right)\cdot 13^{3} + \left(8 a + 2\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 6 + \left(5 a + 4\right)\cdot 13 + 4\cdot 13^{2} + \left(2 a + 4\right)\cdot 13^{3} + a\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 11 + \left(2 a + 3\right)\cdot 13 + \left(3 a + 7\right)\cdot 13^{2} + \left(2 a + 6\right)\cdot 13^{3} + 4 a\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 6\cdot 13 + 7\cdot 13^{2} + 3\cdot 13^{3} + 10\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 1 + \left(7 a + 2\right)\cdot 13 + \left(12 a + 12\right)\cdot 13^{2} + \left(10 a + 5\right)\cdot 13^{3} + \left(11 a + 12\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
