Basic invariants
Dimension: | $3$ |
Group: | $A_4\times C_2$ |
Conductor: | \(10339\)\(\medspace = 7^{2} \cdot 211 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.4.506611.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $A_4\times C_2$ |
Parity: | odd |
Determinant: | 1.211.2t1.a.a |
Projective image: | $A_4$ |
Projective stem field: | Galois closure of 4.0.2181529.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 3x^{4} + 4x^{3} - 6x^{2} + 3x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 38 + 6\cdot 43 + 42\cdot 43^{3} + 39\cdot 43^{4} + 32\cdot 43^{5} +O(43^{6})\) |
$r_{ 2 }$ | $=$ | \( 40 + 24\cdot 43 + 43^{2} + 21\cdot 43^{3} + 29\cdot 43^{4} + 15\cdot 43^{5} +O(43^{6})\) |
$r_{ 3 }$ | $=$ | \( 7 a + 32 + \left(39 a + 13\right)\cdot 43 + \left(19 a + 12\right)\cdot 43^{2} + 12\cdot 43^{3} + \left(14 a + 1\right)\cdot 43^{4} + \left(7 a + 27\right)\cdot 43^{5} +O(43^{6})\) |
$r_{ 4 }$ | $=$ | \( 36 a + 39 + \left(3 a + 2\right)\cdot 43 + \left(23 a + 36\right)\cdot 43^{2} + \left(42 a + 35\right)\cdot 43^{3} + \left(28 a + 14\right)\cdot 43^{4} + \left(35 a + 20\right)\cdot 43^{5} +O(43^{6})\) |
$r_{ 5 }$ | $=$ | \( 24 a + \left(33 a + 14\right)\cdot 43 + 25 a\cdot 43^{2} + \left(a + 21\right)\cdot 43^{3} + \left(37 a + 25\right)\cdot 43^{4} + \left(a + 12\right)\cdot 43^{5} +O(43^{6})\) |
$r_{ 6 }$ | $=$ | \( 19 a + 24 + \left(9 a + 23\right)\cdot 43 + \left(17 a + 35\right)\cdot 43^{2} + \left(41 a + 39\right)\cdot 43^{3} + \left(5 a + 17\right)\cdot 43^{4} + \left(41 a + 20\right)\cdot 43^{5} +O(43^{6})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$3$ | $2$ | $(1,2)$ | $1$ |
$3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$4$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
$4$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
$4$ | $6$ | $(1,6,4,2,5,3)$ | $0$ |
$4$ | $6$ | $(1,3,5,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.