Basic invariants
Dimension: | $2$ |
Group: | $D_{6}$ |
Conductor: | \(176\)\(\medspace = 2^{4} \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.30976.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{6}$ |
Parity: | odd |
Determinant: | 1.11.2t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.44.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 2x^{4} - 2x^{3} + 4x^{2} - 4x + 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 13 + 5\cdot 13^{2} + 8\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$ |
$=$ |
\( 7 a + 5 + \left(8 a + 6\right)\cdot 13 + \left(a + 1\right)\cdot 13^{2} + \left(4 a + 5\right)\cdot 13^{3} + \left(12 a + 7\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 9 a + 5 + \left(12 a + 4\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(11 a + 7\right)\cdot 13^{3} + \left(a + 11\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 + 8\cdot 13 + 2\cdot 13^{2} + 6\cdot 13^{3} + 7\cdot 13^{4} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a + 1 + 8\cdot 13 + \left(7 a + 7\right)\cdot 13^{2} + \left(a + 12\right)\cdot 13^{3} + \left(11 a + 1\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 6 a + 12 + \left(4 a + 7\right)\cdot 13 + \left(11 a + 7\right)\cdot 13^{2} + \left(8 a + 7\right)\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
$3$ | $2$ | $(1,2)(4,5)$ | $0$ |
$3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
$2$ | $3$ | $(1,6,2)(3,5,4)$ | $-1$ |
$2$ | $6$ | $(1,5,6,4,2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.