Galois orbit of Artin representations 3.2e2_3e3_7e2_23e2.6t11.1

• Label: 3.2e2_3e3_7e2_23e2.6t11.1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 23^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$2799468= 2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{6} - 3 x^{5} + 2 x^{4} + x^{3} + 6 x^{2} - 7 x + 3 $ 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 $\Q_{ 73 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 64 a + 5 + \left(14 a + 5\right)\cdot 73 + \left(28 a + 51\right)\cdot 73^{2} + \left(62 a + 28\right)\cdot 73^{3} + 15\cdot 73^{4} + \left(3 a + 11\right)\cdot 73^{5} + \left(68 a + 31\right)\cdot 73^{6} + \left(49 a + 22\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 9 a + 69 + \left(58 a + 67\right)\cdot 73 + \left(44 a + 21\right)\cdot 73^{2} + \left(10 a + 44\right)\cdot 73^{3} + \left(72 a + 57\right)\cdot 73^{4} + \left(69 a + 61\right)\cdot 73^{5} + \left(4 a + 41\right)\cdot 73^{6} + \left(23 a + 50\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 64 a + 23 + \left(14 a + 14\right)\cdot 73 + \left(28 a + 25\right)\cdot 73^{2} + \left(62 a + 31\right)\cdot 73^{3} + 44\cdot 73^{4} + \left(3 a + 53\right)\cdot 73^{5} + \left(68 a + 59\right)\cdot 73^{6} + \left(49 a + 41\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 48 + 71\cdot 73 + 70\cdot 73^{2} + 6\cdot 73^{3} + 28\cdot 73^{4} + 40\cdot 73^{5} + 58\cdot 73^{6} + 5\cdot 73^{7} +O\left(73^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 9 a + 51 + \left(58 a + 58\right)\cdot 73 + \left(44 a + 47\right)\cdot 73^{2} + \left(10 a + 41\right)\cdot 73^{3} + \left(72 a + 28\right)\cdot 73^{4} + \left(69 a + 19\right)\cdot 73^{5} + \left(4 a + 13\right)\cdot 73^{6} + \left(23 a + 31\right)\cdot 73^{7} +O\left(73^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 26 + 73 + 2\cdot 73^{2} + 66\cdot 73^{3} + 44\cdot 73^{4} + 32\cdot 73^{5} + 14\cdot 73^{6} + 67\cdot 73^{7} +O\left(73^{ 8 }\right)$

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

Cycle notation

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

$(4,6)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.