Properties

 Label 4.2e6_3e2_31.6t13.1c1 Dimension 4 Group $C_3^2:D_4$ Conductor $2^{6} \cdot 3^{2} \cdot 31$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $17856= 2^{6} \cdot 3^{2} \cdot 31$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.2e2_31.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{2} + 42 x + 3$
Roots:
 $r_{ 1 }$ $=$ $28 + 29\cdot 43 + 15\cdot 43^{2} + 16\cdot 43^{3} + 4\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 a + 27 + \left(29 a + 8\right)\cdot 43 + \left(33 a + 27\right)\cdot 43^{2} + \left(21 a + 13\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $41 a + 9 + \left(39 a + 7\right)\cdot 43 + \left(37 a + 36\right)\cdot 43^{2} + \left(25 a + 40\right)\cdot 43^{3} + \left(33 a + 36\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $41 a + 29 + \left(13 a + 35\right)\cdot 43 + \left(9 a + 31\right)\cdot 43^{2} + \left(21 a + 1\right)\cdot 43^{3} + \left(4 a + 19\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 5 }$ $=$ $31 + 41\cdot 43 + 26\cdot 43^{2} + 27\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 6 }$ $=$ $2 a + 7 + \left(3 a + 6\right)\cdot 43 + \left(5 a + 34\right)\cdot 43^{2} + \left(17 a + 28\right)\cdot 43^{3} + \left(9 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2)(3,4)(5,6)$ $(1,3)$ $(1,3,6)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,6)$ $2$ $9$ $2$ $(3,6)(4,5)$ $0$ $4$ $3$ $(1,3,6)$ $1$ $4$ $3$ $(1,3,6)(2,4,5)$ $-2$ $18$ $4$ $(1,2)(3,5,6,4)$ $0$ $12$ $6$ $(1,4,3,5,6,2)$ $0$ $12$ $6$ $(2,4,5)(3,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.