Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(5751\)\(\medspace = 3^{4} \cdot 71 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.465831.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.71.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.4.408321.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 9x^{4} - 13x^{3} + 15x^{2} - 9x + 3 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 11 + \left(8 a + 13\right)\cdot 19 + \left(a + 3\right)\cdot 19^{2} + \left(3 a + 18\right)\cdot 19^{3} + \left(9 a + 15\right)\cdot 19^{4} + 13\cdot 19^{5} + \left(14 a + 2\right)\cdot 19^{6} + \left(2 a + 15\right)\cdot 19^{7} + a\cdot 19^{8} + \left(3 a + 18\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 9 + \left(10 a + 5\right)\cdot 19 + \left(17 a + 15\right)\cdot 19^{2} + 15 a\cdot 19^{3} + \left(9 a + 3\right)\cdot 19^{4} + \left(18 a + 5\right)\cdot 19^{5} + \left(4 a + 16\right)\cdot 19^{6} + \left(16 a + 3\right)\cdot 19^{7} + \left(17 a + 18\right)\cdot 19^{8} + 15 a\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 8\cdot 19 + 7\cdot 19^{2} + 11\cdot 19^{3} + 13\cdot 19^{4} + 8\cdot 19^{5} + 7\cdot 19^{6} + 4\cdot 19^{7} + 19^{8} + 3\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 8 + 11\cdot 19 + \left(a + 18\right)\cdot 19^{2} + \left(11 a + 13\right)\cdot 19^{3} + \left(11 a + 18\right)\cdot 19^{4} + \left(18 a + 5\right)\cdot 19^{5} + 18\cdot 19^{6} + \left(11 a + 13\right)\cdot 19^{7} + \left(18 a + 5\right)\cdot 19^{8} + \left(13 a + 2\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 12 + \left(18 a + 7\right)\cdot 19 + 17 a\cdot 19^{2} + \left(7 a + 5\right)\cdot 19^{3} + 7 a\cdot 19^{4} + 13\cdot 19^{5} + 18 a\cdot 19^{6} + \left(7 a + 5\right)\cdot 19^{7} + 13\cdot 19^{8} + \left(5 a + 16\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 + 10\cdot 19 + 11\cdot 19^{2} + 7\cdot 19^{3} + 5\cdot 19^{4} + 10\cdot 19^{5} + 11\cdot 19^{6} + 14\cdot 19^{7} + 17\cdot 19^{8} + 15\cdot 19^{9} +O(19^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $4$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,2,4,3)$ | $0$ |
| $4$ | $6$ | $(1,3,4,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.