Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(8667\)\(\medspace = 3^{4} \cdot 107 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.702027.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Determinant: | 1.107.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.927369.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 3x^{5} + 3x^{4} - 9x^{2} + 6x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 4 a + \left(7 a + 6\right)\cdot 19 + \left(14 a + 3\right)\cdot 19^{2} + \left(2 a + 10\right)\cdot 19^{3} + \left(3 a + 8\right)\cdot 19^{5} + \left(18 a + 12\right)\cdot 19^{6} +O(19^{7})\)
$r_{ 2 }$ |
$=$ |
\( 15 a + 4 + \left(11 a + 9\right)\cdot 19 + \left(4 a + 10\right)\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(18 a + 16\right)\cdot 19^{4} + \left(15 a + 10\right)\cdot 19^{5} + 8\cdot 19^{6} +O(19^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 13 + 3\cdot 19 + 15\cdot 19^{2} + 9\cdot 19^{3} + 4\cdot 19^{4} + 5\cdot 19^{5} + 15\cdot 19^{6} +O(19^{7})\)
| $r_{ 4 }$ |
$=$ |
\( 14 + 10\cdot 19 + 7\cdot 19^{2} + 7\cdot 19^{3} + 17\cdot 19^{4} + 12\cdot 19^{5} + 17\cdot 19^{6} +O(19^{7})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a + 13 + 5\cdot 19 + \left(3 a + 18\right)\cdot 19^{2} + \left(a + 6\right)\cdot 19^{3} + \left(11 a + 13\right)\cdot 19^{4} + \left(8 a + 1\right)\cdot 19^{5} + \left(a + 5\right)\cdot 19^{6} +O(19^{7})\)
| $r_{ 6 }$ |
$=$ |
\( 16 a + 16 + \left(18 a + 2\right)\cdot 19 + \left(15 a + 2\right)\cdot 19^{2} + \left(17 a + 5\right)\cdot 19^{3} + \left(7 a + 4\right)\cdot 19^{4} + \left(10 a + 18\right)\cdot 19^{5} + \left(17 a + 16\right)\cdot 19^{6} +O(19^{7})\)
| |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$3$ | $2$ | $(1,2)$ | $1$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
$4$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
$4$ | $6$ | $(1,6,4,2,5,3)$ | $0$ |
$4$ | $6$ | $(1,3,5,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.