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