Properties

Label 6.41e3_257e4.12t111.1c1
Dimension 6
Group $V_4^2:(S_3\times C_2)$
Conductor $ 41^{3} \cdot 257^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$6$
Group:$V_4^2:(S_3\times C_2)$
Conductor:$300665822507321= 41^{3} \cdot 257^{4} $
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: 12T111
Parity: Even
Determinant: 1.41.2t1.1c1

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

SizeOrderAction 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$$(1,3)(2,4)(5,6)(7,8)$$0$
$12$$2$$(1,2)(3,4)$$-2$
$32$$3$$(2,5,7)(3,4,6)$$0$
$12$$4$$(1,8,5,3)(2,4,7,6)$$0$
$12$$4$$(1,3,5,8)(2,4,7,6)$$0$
$12$$4$$(1,5,2,7)(3,6,4,8)$$2$
$24$$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.