# Properties

 Label 6.41e3_257e3.12t108.1c1 Dimension 6 Group $V_4^2:(S_3\times C_2)$ Conductor $41^{3} \cdot 257^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $V_4^2:(S_3\times C_2)$ Conductor: $1169905924153= 41^{3} \cdot 257^{3}$ Artin number field: Splitting field of $f=x^{8} - x^{7} - 8 x^{6} + 6 x^{5} + 17 x^{4} - 13 x^{3} + x^{2} + 17 x - 10$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 12T108 Parity: Even Determinant: 1.41_257.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 36.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{3} + 2 x + 18$
Roots: \begin{aligned} r_{ 1 } &= 366885921668117010924336599577951894907708006638 a^{2} + 2076229504335802977811363721205101015518829261562 a + 5183203666960888269083325702816570012032927171064 +O\left(23^{ 36 }\right) \\ r_{ 2 } &= 4109870704983461179459259497590345173392090775981 a^{2} + 1939791891119748515332045085474313472633837866968 a + 3157506627265235587878398666097032457129069839414 +O\left(23^{ 36 }\right) \\ r_{ 3 } &= -4689761996566044864345324121331595382801344524699 +O\left(23^{ 36 }\right) \\ r_{ 4 } &= 1896443298826592689545165351948836393945303540930 a^{2} - 4739961160075211928871273427315395565762879666760 a - 1854072527498206474989443209027510496721143089592 +O\left(23^{ 36 }\right) \\ r_{ 5 } &= -4476756626651578190383596097168297068299798782619 a^{2} - 4016021395455551493143408806679414488152667128530 a - 4783158106190094452286330009879792902018599059999 +O\left(23^{ 36 }\right) \\ r_{ 6 } &= 3895207486774258994097070957176064054699646119635 a^{2} - 3617204925083564177348249095998542695319992309665 a - 2697225318959373855212647852425904578823536830039 +O\left(23^{ 36 }\right) \\ r_{ 7 } &= 3755685460156838283941684446071569992278038470556 +O\left(23^{ 36 }\right) \\ r_{ 8 } &= 4732864340573315675235000041979192440679601875596 a^{2} - 2167349041015391252657713827790154628241679559736 a + 1927822194830757505930336377679630898924588023296 +O\left(23^{ 36 }\right) \\ \end{aligned}

### Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(2,7,5)(3,8,6)$ $(5,7)(6,8)$ $(3,8)(4,6)$ $(1,2)(3,8)(4,6)(5,7)$ $(3,4)(6,8)$ $(1,3)(2,4)(5,6)(7,8)$ $(1,7)(2,5)(3,8)(4,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $6$ $3$ $2$ $(1,5)(2,7)(3,8)(4,6)$ $-2$ $4$ $2$ $(1,8)(2,6)(3,5)(4,7)$ $0$ $6$ $2$ $(1,2)(3,8)(4,6)(5,7)$ $2$ $6$ $2$ $(1,5)(2,7)$ $-2$ $12$ $2$ $(1,3)(2,4)(5,6)(7,8)$ $-2$ $12$ $2$ $(1,2)(3,4)$ $0$ $32$ $3$ $(2,5,7)(3,4,6)$ $0$ $12$ $4$ $(1,8,5,3)(2,4,7,6)$ $2$ $12$ $4$ $(1,3,5,8)(2,4,7,6)$ $0$ $12$ $4$ $(1,5,2,7)(3,6,4,8)$ $0$ $24$ $4$ $(1,3,7,8)(2,4,5,6)$ $0$ $24$ $4$ $(3,6,4,8)(5,7)$ $0$ $32$ $6$ $(1,8)(2,4,5,6,7,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.