Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1840\)\(\medspace = 2^{4} \cdot 5 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.3385600.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.115.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.460.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{3} + 16x^{2} - 24x + 18 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 11 + \left(a + 9\right)\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + 5 a\cdot 13^{3} + \left(a + 1\right)\cdot 13^{4} + \left(7 a + 1\right)\cdot 13^{5} + \left(7 a + 11\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 2 }$ |
$=$ |
\( 8 + 4\cdot 13^{2} + 10\cdot 13^{3} + 13^{4} + 5\cdot 13^{5} + 3\cdot 13^{6} +O(13^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 7 a + 10 + \left(11 a + 12\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + \left(8 a + 8\right)\cdot 13^{3} + \left(9 a + 12\right)\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(9 a + 2\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 12 + 8\cdot 13 + 7\cdot 13^{2} + 5\cdot 13^{3} + 12\cdot 13^{4} + 2\cdot 13^{5} + 2\cdot 13^{6} +O(13^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 4 a + 7 + \left(11 a + 2\right)\cdot 13 + \left(9 a + 5\right)\cdot 13^{2} + \left(7 a + 2\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} + \left(5 a + 6\right)\cdot 13^{5} + \left(5 a + 11\right)\cdot 13^{6} +O(13^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 6 a + 4 + \left(a + 4\right)\cdot 13 + \left(7 a + 6\right)\cdot 13^{2} + \left(4 a + 11\right)\cdot 13^{3} + 3 a\cdot 13^{4} + \left(10 a + 8\right)\cdot 13^{5} + \left(3 a + 8\right)\cdot 13^{6} +O(13^{7})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,2,5)(3,4,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,5,3,2,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.