Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(5239\)\(\medspace = 13^{2} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.885391.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.31.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.162409.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 3x^{4} + 7x^{3} - x^{2} - 3x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 43 + 26\cdot 47 + 25\cdot 47^{2} + 22\cdot 47^{3} + 19\cdot 47^{4} + 44\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 a + 38 + \left(38 a + 8\right)\cdot 47 + \left(13 a + 19\right)\cdot 47^{2} + \left(23 a + 39\right)\cdot 47^{3} + 14 a\cdot 47^{4} + \left(45 a + 35\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 18 + \left(8 a + 2\right)\cdot 47 + \left(33 a + 8\right)\cdot 47^{2} + \left(23 a + 25\right)\cdot 47^{3} + \left(32 a + 6\right)\cdot 47^{4} + \left(a + 17\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 42 + \left(25 a + 45\right)\cdot 47 + \left(16 a + 15\right)\cdot 47^{2} + \left(9 a + 38\right)\cdot 47^{3} + \left(38 a + 42\right)\cdot 47^{4} + \left(24 a + 42\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a + 19 + \left(21 a + 37\right)\cdot 47 + \left(30 a + 23\right)\cdot 47^{2} + \left(37 a + 40\right)\cdot 47^{3} + \left(8 a + 15\right)\cdot 47^{4} + \left(22 a + 7\right)\cdot 47^{5} +O(47^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 19\cdot 47 + 47^{2} + 22\cdot 47^{3} + 8\cdot 47^{4} + 41\cdot 47^{5} +O(47^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,3)$ | $-1$ |
| $4$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $4$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $4$ | $6$ | $(1,2,4,6,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.