Basic invariants
Dimension: | $4$ |
Group: | $C_3^2:D_4$ |
Conductor: | \(43754283\)\(\medspace = 3^{3} \cdot 19^{2} \cdot 67^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.34371.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T34 |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $\SOPlus(4,2)$ |
Projective stem field: | Galois closure of 6.0.34371.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 2x^{3} + 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 9 a + 23 + \left(15 a + 19\right)\cdot 31 + \left(21 a + 17\right)\cdot 31^{2} + \left(26 a + 14\right)\cdot 31^{3} + \left(a + 22\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 22 a + 10 + \left(15 a + 10\right)\cdot 31 + \left(9 a + 14\right)\cdot 31^{2} + \left(4 a + 15\right)\cdot 31^{3} + \left(29 a + 30\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 24 + 17\cdot 31 + 5\cdot 31^{2} + 23\cdot 31^{3} + 21\cdot 31^{4} +O(31^{5})\) |
$r_{ 4 }$ | $=$ | \( 2 a + 25 + \left(11 a + 15\right)\cdot 31 + \left(9 a + 21\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(11 a + 28\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 5 }$ | $=$ | \( 14 + 24\cdot 31 + 4\cdot 31^{2} + 25\cdot 31^{3} + 24\cdot 31^{4} +O(31^{5})\) |
$r_{ 6 }$ | $=$ | \( 29 a + 29 + \left(19 a + 4\right)\cdot 31 + \left(21 a + 29\right)\cdot 31^{2} + \left(5 a + 27\right)\cdot 31^{3} + \left(19 a + 26\right)\cdot 31^{4} +O(31^{5})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$6$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
$6$ | $2$ | $(2,3)$ | $0$ |
$9$ | $2$ | $(2,3)(5,6)$ | $0$ |
$4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$4$ | $3$ | $(1,2,3)$ | $-2$ |
$18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
$12$ | $6$ | $(1,5,2,6,3,4)$ | $1$ |
$12$ | $6$ | $(2,3)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.