Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(981\)\(\medspace = 3^{2} \cdot 109 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.320787.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.109.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2943.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} + x^{3} + 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 36 + \left(40 a + 13\right)\cdot 67 + \left(31 a + 15\right)\cdot 67^{2} + \left(16 a + 39\right)\cdot 67^{3} + \left(40 a + 23\right)\cdot 67^{4} + \left(43 a + 4\right)\cdot 67^{5} + \left(38 a + 13\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 50 + \left(24 a + 40\right)\cdot 67 + \left(24 a + 63\right)\cdot 67^{2} + \left(57 a + 15\right)\cdot 67^{3} + \left(57 a + 21\right)\cdot 67^{4} + \left(30 a + 36\right)\cdot 67^{5} + \left(7 a + 33\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 50 + 54\cdot 67 + 59\cdot 67^{2} + 62\cdot 67^{3} + 60\cdot 67^{4} + 33\cdot 67^{5} + 26\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 + 61\cdot 67 + 24\cdot 67^{2} + 56\cdot 67^{3} + 66\cdot 67^{4} + 19\cdot 67^{5} + 38\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 60 a + 64 + \left(26 a + 32\right)\cdot 67 + \left(35 a + 34\right)\cdot 67^{2} + \left(50 a + 6\right)\cdot 67^{3} + \left(26 a + 34\right)\cdot 67^{4} + \left(23 a + 4\right)\cdot 67^{5} + \left(28 a + 57\right)\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 a + 7 + \left(42 a + 64\right)\cdot 67 + \left(42 a + 2\right)\cdot 67^{2} + \left(9 a + 20\right)\cdot 67^{3} + \left(9 a + 61\right)\cdot 67^{4} + \left(36 a + 34\right)\cdot 67^{5} + \left(59 a + 32\right)\cdot 67^{6} +O(67^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,2)$ | $1$ |
| $6$ | $4$ | $(1,6)(2,4,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.