Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(20339\)\(\medspace = 11 \cdot 43^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.37606811.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.4.223729.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 6x^{3} - 4x^{2} - 32x + 64 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 21 + \left(40 a + 19\right)\cdot 47 + \left(13 a + 41\right)\cdot 47^{2} + \left(37 a + 24\right)\cdot 47^{3} + \left(14 a + 14\right)\cdot 47^{4} + \left(7 a + 18\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 a + 43 + \left(6 a + 41\right)\cdot 47 + \left(33 a + 28\right)\cdot 47^{2} + \left(9 a + 38\right)\cdot 47^{3} + \left(32 a + 6\right)\cdot 47^{4} + \left(39 a + 18\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 22\cdot 47 + 26\cdot 47^{2} + 17\cdot 47^{3} + 40\cdot 47^{4} + 16\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 a + 6 + \left(21 a + 14\right)\cdot 47 + \left(42 a + 9\right)\cdot 47^{2} + \left(6 a + 25\right)\cdot 47^{3} + \left(11 a + 39\right)\cdot 47^{4} + \left(8 a + 11\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 35 + \left(25 a + 19\right)\cdot 47 + \left(4 a + 25\right)\cdot 47^{2} + \left(40 a + 43\right)\cdot 47^{3} + \left(35 a + 7\right)\cdot 47^{4} + \left(38 a + 17\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 + 23\cdot 47 + 9\cdot 47^{2} + 38\cdot 47^{3} + 31\cdot 47^{4} + 11\cdot 47^{5} +O(47^{6})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ | ✓ |
| $3$ | $2$ | $(4,5)$ | $1$ | |
| $3$ | $2$ | $(1,2)(4,5)$ | $-1$ | |
| $4$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ | |
| $4$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ | |
| $4$ | $6$ | $(1,4,6,2,5,3)$ | $0$ | |
| $4$ | $6$ | $(1,3,5,2,6,4)$ | $0$ |