Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3120\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.29203200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.195.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.780.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 8x^{4} - 18x^{3} + 4x^{2} + 12x + 6 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 21 + \left(13 a + 1\right)\cdot 47 + \left(31 a + 24\right)\cdot 47^{2} + \left(14 a + 24\right)\cdot 47^{3} + \left(29 a + 26\right)\cdot 47^{4} + \left(40 a + 18\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 a + 12 + \left(2 a + 35\right)\cdot 47 + \left(26 a + 22\right)\cdot 47^{2} + \left(9 a + 44\right)\cdot 47^{3} + \left(26 a + 38\right)\cdot 47^{4} + \left(18 a + 21\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 a + 4 + \left(33 a + 13\right)\cdot 47 + \left(15 a + 26\right)\cdot 47^{2} + \left(32 a + 22\right)\cdot 47^{3} + \left(17 a + 23\right)\cdot 47^{4} + \left(6 a + 23\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 32\cdot 47 + 43\cdot 47^{2} + 46\cdot 47^{3} + 43\cdot 47^{4} + 4\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 9 + \left(44 a + 18\right)\cdot 47 + \left(20 a + 25\right)\cdot 47^{2} + \left(37 a + 37\right)\cdot 47^{3} + \left(20 a + 34\right)\cdot 47^{4} + \left(28 a + 32\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 + 40\cdot 47 + 45\cdot 47^{2} + 11\cdot 47^{3} + 20\cdot 47^{4} + 39\cdot 47^{5} +O(47^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,2,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.