Galois orbit of Artin representations 2.140.6t5.a

• Label: 2.140.6t5.a
• Dimension: $2$
• Group: $S_3\times C_3$
• Conductor: $140$
• Indicator: $0$

Basic invariants

Dimension:	$2$
Group:	$S_3\times C_3$
Conductor:	$140$$\medspace = 2^{2} \cdot 5 \cdot 7$
Artin number field:	Galois closure of 6.0.392000.1
Galois orbit size:	$2$
Smallest permutation container:	$S_3\times C_3$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.980.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12x + 2$

Roots:

$r_{ 1 }$	$=$	$12 a + 6 + \left(7 a + 4\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(7 a + 3\right)\cdot 13^{4} + \left(11 a + 10\right)\cdot 13^{5} + \left(6 a + 2\right)\cdot 13^{6} +O(13^{7})$
$r_{ 2 }$	$=$	$7 a + 11 + \left(10 a + 8\right)\cdot 13 + \left(3 a + 8\right)\cdot 13^{2} + 9\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + \left(2 a + 7\right)\cdot 13^{5} + \left(12 a + 12\right)\cdot 13^{6} +O(13^{7})$
$r_{ 3 }$	$=$	$a + 5 + 5 a\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(5 a + 1\right)\cdot 13^{4} + \left(a + 1\right)\cdot 13^{5} + \left(6 a + 11\right)\cdot 13^{6} +O(13^{7})$
$r_{ 4 }$	$=$	$6 a + 5 + \left(2 a + 12\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(12 a + 6\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} + \left(10 a + 6\right)\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})$
$r_{ 5 }$	$=$	$8 a + 3 + \left(2 a + 9\right)\cdot 13 + \left(7 a + 9\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} + \left(3 a + 8\right)\cdot 13^{5} + 5 a\cdot 13^{6} +O(13^{7})$
$r_{ 6 }$	$=$	$5 a + 11 + \left(10 a + 3\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(4 a + 3\right)\cdot 13^{4} + \left(9 a + 4\right)\cdot 13^{5} + \left(7 a + 2\right)\cdot 13^{6} +O(13^{7})$

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

Cycle notation

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

$(2,6,3)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$3$	$2$	$(1,3)(2,4)(5,6)$	$0$	$0$
$1$	$3$	$(1,4,5)(2,6,3)$	$2 \zeta_{3}$	$-2 \zeta_{3} - 2$
$1$	$3$	$(1,5,4)(2,3,6)$	$-2 \zeta_{3} - 2$	$2 \zeta_{3}$
$2$	$3$	$(2,6,3)$	$\zeta_{3} + 1$	$-\zeta_{3}$
$2$	$3$	$(2,3,6)$	$-\zeta_{3}$	$\zeta_{3} + 1$
$2$	$3$	$(1,4,5)(2,3,6)$	$-1$	$-1$
$3$	$6$	$(1,2,4,6,5,3)$	$0$	$0$
$3$	$6$	$(1,3,5,6,4,2)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.