Basic invariants
| Dimension: | $4$ |
| Group: | $S_3^2$ |
| Conductor: | \(63504\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.1778112.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3^2$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3^2$ |
| Projective stem field: | Galois closure of 6.2.1778112.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} - x^{2} - 2x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 4 + \left(17 a + 3\right)\cdot 19 + \left(2 a + 13\right)\cdot 19^{2} + \left(6 a + 3\right)\cdot 19^{3} + \left(16 a + 10\right)\cdot 19^{4} + \left(17 a + 13\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 1 + \left(2 a + 13\right)\cdot 19 + \left(11 a + 1\right)\cdot 19^{2} + \left(9 a + 1\right)\cdot 19^{3} + 12 a\cdot 19^{4} + \left(5 a + 5\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 2\cdot 19 + 14\cdot 19^{3} + 15\cdot 19^{4} + 6\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 2\cdot 19 + 14\cdot 19^{2} + 12\cdot 19^{3} + 7\cdot 19^{4} + 18\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 13 + \left(a + 11\right)\cdot 19 + \left(16 a + 17\right)\cdot 19^{2} + \left(12 a + 6\right)\cdot 19^{3} + \left(2 a + 1\right)\cdot 19^{4} + \left(a + 15\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 12 + \left(16 a + 4\right)\cdot 19 + \left(7 a + 10\right)\cdot 19^{2} + \left(9 a + 18\right)\cdot 19^{3} + \left(6 a + 2\right)\cdot 19^{4} + \left(13 a + 17\right)\cdot 19^{5} +O(19^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $9$ | $2$ | $(2,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,4,6)$ | $-2$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,5,3)$ | $1$ |
| $6$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |
| $6$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.