Galois orbit of Artin representations 3.3_307e2.6t11.2

• Label: 3.3_307e2.6t11.2
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 3 \cdot 307^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$282747= 3 \cdot 307^{2} $
Artin number field:	Splitting field of $f= x^{6} - 3 x^{5} + 2 x^{4} + x^{3} + 2 x^{2} - 3 x + 2 $ 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 7.

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

Roots:

$r_{ 1 }$	$=$	$ 41 + 40\cdot 73 + 49\cdot 73^{2} + 16\cdot 73^{3} + 41\cdot 73^{4} + 68\cdot 73^{5} + 40\cdot 73^{6} +O\left(73^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 64 a + 44 + \left(7 a + 55\right)\cdot 73 + \left(14 a + 26\right)\cdot 73^{2} + \left(53 a + 62\right)\cdot 73^{3} + \left(9 a + 13\right)\cdot 73^{4} + \left(52 a + 59\right)\cdot 73^{5} + \left(37 a + 26\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 9 a + 30 + \left(65 a + 17\right)\cdot 73 + \left(58 a + 46\right)\cdot 73^{2} + \left(19 a + 10\right)\cdot 73^{3} + \left(63 a + 59\right)\cdot 73^{4} + \left(20 a + 13\right)\cdot 73^{5} + \left(35 a + 46\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 33 + 32\cdot 73 + 23\cdot 73^{2} + 56\cdot 73^{3} + 31\cdot 73^{4} + 4\cdot 73^{5} + 32\cdot 73^{6} +O\left(73^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 64 a + 57 + \left(7 a + 57\right)\cdot 73 + \left(14 a + 11\right)\cdot 73^{2} + \left(53 a + 11\right)\cdot 73^{3} + \left(9 a + 10\right)\cdot 73^{4} + \left(52 a + 13\right)\cdot 73^{5} + \left(37 a + 58\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 9 a + 17 + \left(65 a + 15\right)\cdot 73 + \left(58 a + 61\right)\cdot 73^{2} + \left(19 a + 61\right)\cdot 73^{3} + \left(63 a + 62\right)\cdot 73^{4} + \left(20 a + 59\right)\cdot 73^{5} + \left(35 a + 14\right)\cdot 73^{6} +O\left(73^{ 7 }\right)$

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

Cycle notation

$(1,4)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.