Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(33856\)\(\medspace = 2^{6} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.33856.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 6.2.33856.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + x^{4} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 13\cdot 19 + 5\cdot 19^{2} + 3\cdot 19^{3} + 18\cdot 19^{4} + 5\cdot 19^{5} + 2\cdot 19^{6} + 16\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a + 4 + \left(7 a + 3\right)\cdot 19 + \left(8 a + 14\right)\cdot 19^{2} + \left(14 a + 8\right)\cdot 19^{3} + \left(12 a + 6\right)\cdot 19^{4} + \left(10 a + 7\right)\cdot 19^{5} + \left(11 a + 6\right)\cdot 19^{6} + \left(15 a + 2\right)\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 16 + \left(7 a + 6\right)\cdot 19 + \left(8 a + 4\right)\cdot 19^{2} + \left(14 a + 4\right)\cdot 19^{3} + \left(12 a + 14\right)\cdot 19^{4} + \left(10 a + 13\right)\cdot 19^{5} + \left(11 a + 11\right)\cdot 19^{6} + \left(15 a + 12\right)\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 5\cdot 19 + 13\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{5} + 16\cdot 19^{6} + 2\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 15 + \left(11 a + 15\right)\cdot 19 + \left(10 a + 4\right)\cdot 19^{2} + \left(4 a + 10\right)\cdot 19^{3} + \left(6 a + 12\right)\cdot 19^{4} + \left(8 a + 11\right)\cdot 19^{5} + \left(7 a + 12\right)\cdot 19^{6} + \left(3 a + 16\right)\cdot 19^{7} +O(19^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 3 + \left(11 a + 12\right)\cdot 19 + \left(10 a + 14\right)\cdot 19^{2} + \left(4 a + 14\right)\cdot 19^{3} + \left(6 a + 4\right)\cdot 19^{4} + \left(8 a + 5\right)\cdot 19^{5} + \left(7 a + 7\right)\cdot 19^{6} + \left(3 a + 6\right)\cdot 19^{7} +O(19^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.