Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(3303\)\(\medspace = 3^{2} \cdot 367 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.3636603.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.367.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.4.9909.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 8x^{4} + 12x^{2} + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 2\cdot 31 + 4\cdot 31^{2} + 7\cdot 31^{3} + 8\cdot 31^{4} + 22\cdot 31^{5} + 18\cdot 31^{7} + 9\cdot 31^{8} + 29\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 19 + \left(8 a + 28\right)\cdot 31 + \left(25 a + 9\right)\cdot 31^{2} + \left(17 a + 10\right)\cdot 31^{3} + \left(30 a + 9\right)\cdot 31^{4} + \left(21 a + 24\right)\cdot 31^{5} + \left(3 a + 22\right)\cdot 31^{6} + \left(3 a + 29\right)\cdot 31^{7} + \left(12 a + 4\right)\cdot 31^{8} + \left(8 a + 13\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 29 + \left(23 a + 8\right)\cdot 31 + \left(28 a + 29\right)\cdot 31^{2} + \left(15 a + 13\right)\cdot 31^{3} + \left(27 a + 11\right)\cdot 31^{4} + \left(30 a + 29\right)\cdot 31^{5} + \left(4 a + 25\right)\cdot 31^{6} + \left(27 a + 21\right)\cdot 31^{7} + 12 a\cdot 31^{8} + \left(11 a + 26\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 + 28\cdot 31 + 26\cdot 31^{2} + 23\cdot 31^{3} + 22\cdot 31^{4} + 8\cdot 31^{5} + 30\cdot 31^{6} + 12\cdot 31^{7} + 21\cdot 31^{8} + 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 12 + \left(22 a + 2\right)\cdot 31 + \left(5 a + 21\right)\cdot 31^{2} + \left(13 a + 20\right)\cdot 31^{3} + 21\cdot 31^{4} + \left(9 a + 6\right)\cdot 31^{5} + \left(27 a + 8\right)\cdot 31^{6} + \left(27 a + 1\right)\cdot 31^{7} + \left(18 a + 26\right)\cdot 31^{8} + \left(22 a + 17\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a + 2 + \left(7 a + 22\right)\cdot 31 + \left(2 a + 1\right)\cdot 31^{2} + \left(15 a + 17\right)\cdot 31^{3} + \left(3 a + 19\right)\cdot 31^{4} + 31^{5} + \left(26 a + 5\right)\cdot 31^{6} + \left(3 a + 9\right)\cdot 31^{7} + \left(18 a + 30\right)\cdot 31^{8} + \left(19 a + 4\right)\cdot 31^{9} +O(31^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $-1$ |
| $8$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.