Properties

 Label 4.7e2_317.6t13.1 Dimension 4 Group $C_3^2:D_4$ Conductor $7^{2} \cdot 317$ Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $15533= 7^{2} \cdot 317$ Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - 2 x^{3} + x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $18 a + 15 + \left(8 a + 18\right)\cdot 23 + \left(a + 8\right)\cdot 23^{2} + \left(16 a + 12\right)\cdot 23^{3} + \left(a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 18\cdot 23 + 2\cdot 23^{2} + 8\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $a + 4 + \left(11 a + 14\right)\cdot 23 + 20\cdot 23^{2} + \left(3 a + 5\right)\cdot 23^{3} + 10 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $5 a + 5 + \left(14 a + 18\right)\cdot 23 + \left(21 a + 2\right)\cdot 23^{2} + \left(6 a + 20\right)\cdot 23^{3} + \left(21 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $22 a + 6 + \left(11 a + 12\right)\cdot 23 + \left(22 a + 10\right)\cdot 23^{2} + \left(19 a + 11\right)\cdot 23^{3} + \left(12 a + 17\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 6 }$ $=$ $13 + 10\cdot 23 + 11\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $4$ $3$ $(1,4,6)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.