Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{2} + 12 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 2\cdot 13 + 6\cdot 13^{2} + 11\cdot 13^{3} + 7\cdot 13^{4} + 13^{6} + 9\cdot 13^{7} + 3\cdot 13^{8} + 5\cdot 13^{9} + 3\cdot 13^{10} +O\left(13^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 4 + \left(12 a + 7\right)\cdot 13 + \left(6 a + 7\right)\cdot 13^{2} + \left(12 a + 2\right)\cdot 13^{3} + \left(a + 8\right)\cdot 13^{4} + \left(8 a + 4\right)\cdot 13^{5} + \left(4 a + 1\right)\cdot 13^{6} + \left(2 a + 11\right)\cdot 13^{7} + \left(10 a + 5\right)\cdot 13^{8} + \left(a + 7\right)\cdot 13^{9} +O\left(13^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 6 + \left(3 a + 10\right)\cdot 13 + \left(3 a + 4\right)\cdot 13^{2} + \left(6 a + 2\right)\cdot 13^{3} + \left(6 a + 8\right)\cdot 13^{4} + \left(2 a + 3\right)\cdot 13^{5} + \left(a + 6\right)\cdot 13^{6} + \left(2 a + 10\right)\cdot 13^{7} + \left(7 a + 7\right)\cdot 13^{8} + 3 a\cdot 13^{9} + \left(2 a + 12\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 11 + 12\cdot 13 + \left(6 a + 1\right)\cdot 13^{2} + 8\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} + \left(4 a + 10\right)\cdot 13^{5} + \left(8 a + 10\right)\cdot 13^{6} + \left(10 a + 8\right)\cdot 13^{7} + 2 a\cdot 13^{8} + \left(11 a + 12\right)\cdot 13^{9} + \left(12 a + 11\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 + 7\cdot 13 + 9\cdot 13^{3} + 8\cdot 13^{4} + 6\cdot 13^{5} + 13^{6} + 13^{7} + 8\cdot 13^{8} + 3\cdot 13^{9} +O\left(13^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 8 + \left(9 a + 11\right)\cdot 13 + \left(9 a + 4\right)\cdot 13^{2} + \left(6 a + 5\right)\cdot 13^{3} + \left(6 a + 8\right)\cdot 13^{4} + \left(10 a + 12\right)\cdot 13^{5} + \left(11 a + 4\right)\cdot 13^{6} + \left(10 a + 11\right)\cdot 13^{7} + \left(5 a + 12\right)\cdot 13^{8} + \left(9 a + 9\right)\cdot 13^{9} + \left(10 a + 10\right)\cdot 13^{10} +O\left(13^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(5,6)$ |
| $(1,3,2)(4,6,5)$ |
| $(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(1,5)(2,6)$ |
$-1$ |
| $3$ |
$2$ |
$(2,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,2)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,5,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,5)(2,4,6,3)$ |
$-1$ |
| $8$ |
$6$ |
$(1,3,2,5,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
