Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 6.0.419904.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 6.0.419904.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{3} + 9x^{2} + 6x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 2 + \left(5 a + 10\right)\cdot 11 + \left(8 a + 5\right)\cdot 11^{2} + \left(2 a + 4\right)\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 10 + \left(5 a + 6\right)\cdot 11 + \left(2 a + 1\right)\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 1 + \left(6 a + 5\right)\cdot 11 + \left(6 a + 7\right)\cdot 11^{2} + \left(4 a + 7\right)\cdot 11^{3} + \left(3 a + 5\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 8 + \left(10 a + 6\right)\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + \left(3 a + 9\right)\cdot 11^{3} + \left(7 a + 6\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 10 + 9\cdot 11 + \left(4 a + 3\right)\cdot 11^{2} + \left(7 a + 6\right)\cdot 11^{3} + \left(3 a + 10\right)\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 2 + \left(4 a + 5\right)\cdot 11 + \left(4 a + 5\right)\cdot 11^{2} + \left(6 a + 8\right)\cdot 11^{3} + \left(7 a + 3\right)\cdot 11^{4} +O(11^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,3,4)(2,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.