Galois orbit of Artin representations 3.3e3_5e2_167e2.6t11.1

• Label: 3.3e3_5e2_167e2.6t11.1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 3^{3} \cdot 5^{2} \cdot 167^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$18825075= 3^{3} \cdot 5^{2} \cdot 167^{2} $
Artin number field:	Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{3} - 18 x + 27 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 55 + 2\cdot 67 + 11\cdot 67^{2} + 36\cdot 67^{3} + 48\cdot 67^{4} + 2\cdot 67^{5} + 5\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 25 a + 56 + \left(29 a + 27\right)\cdot 67 + \left(23 a + 26\right)\cdot 67^{2} + \left(57 a + 21\right)\cdot 67^{3} + \left(14 a + 62\right)\cdot 67^{4} + \left(34 a + 24\right)\cdot 67^{5} + \left(61 a + 22\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 42 a + 22 + \left(37 a + 53\right)\cdot 67 + \left(43 a + 23\right)\cdot 67^{2} + \left(9 a + 26\right)\cdot 67^{3} + \left(52 a + 64\right)\cdot 67^{4} + \left(32 a + 12\right)\cdot 67^{5} + \left(5 a + 33\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 50 + 62\cdot 67 + 55\cdot 67^{2} + 42\cdot 67^{3} + 42\cdot 67^{4} + 13\cdot 67^{5} + 35\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 45 a + 54 + \left(65 a + 18\right)\cdot 67 + \left(2 a + 35\right)\cdot 67^{2} + \left(26 a + 53\right)\cdot 67^{3} + 3\cdot 67^{4} + \left(62 a + 50\right)\cdot 67^{5} + \left(34 a + 13\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 22 a + 33 + \left(a + 35\right)\cdot 67 + \left(64 a + 48\right)\cdot 67^{2} + \left(40 a + 20\right)\cdot 67^{3} + \left(66 a + 46\right)\cdot 67^{4} + \left(4 a + 29\right)\cdot 67^{5} + \left(32 a + 24\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$

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

Cycle notation

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

$(1,2)(3,4)$

$(2,3)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$3$
$1$	$2$	$(1,4)(2,3)(5,6)$	$-3$
$3$	$2$	$(2,3)$	$1$
$3$	$2$	$(1,4)(2,3)$	$-1$
$6$	$2$	$(1,5)(4,6)$	$-1$
$6$	$2$	$(1,5)(2,3)(4,6)$	$1$
$8$	$3$	$(1,5,2)(3,4,6)$	$0$
$6$	$4$	$(1,2,4,3)$	$-1$
$6$	$4$	$(1,6,4,5)(2,3)$	$1$
$8$	$6$	$(1,5,2,4,6,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.