Galois orbit of Artin representations 3.7_17_23e2.6t11.2

• Label: 3.7_17_23e2.6t11.2
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 7 \cdot 17 \cdot 23^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$62951= 7 \cdot 17 \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - x^{3} + 4 x^{2} + 2 x + 1 $ 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 $\Q_{ 61 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 18 + 23\cdot 61 + 30\cdot 61^{2} + 47\cdot 61^{3} + 11\cdot 61^{4} + 26\cdot 61^{5} + 38\cdot 61^{6} +O\left(61^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 41 a + 54 + \left(59 a + 34\right)\cdot 61 + \left(56 a + 7\right)\cdot 61^{2} + \left(53 a + 25\right)\cdot 61^{3} + \left(53 a + 56\right)\cdot 61^{4} + \left(2 a + 59\right)\cdot 61^{5} + \left(59 a + 18\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 20 a + 34 + \left(a + 53\right)\cdot 61 + \left(4 a + 4\right)\cdot 61^{2} + \left(7 a + 22\right)\cdot 61^{3} + \left(7 a + 56\right)\cdot 61^{4} + \left(58 a + 8\right)\cdot 61^{5} + \left(a + 14\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 59 a + 26 + \left(9 a + 50\right)\cdot 61 + \left(43 a + 44\right)\cdot 61^{2} + \left(30 a + 60\right)\cdot 61^{3} + \left(34 a + 15\right)\cdot 61^{4} + \left(54 a + 33\right)\cdot 61^{5} + \left(23 a + 5\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 2 a + 24 + \left(51 a + 1\right)\cdot 61 + \left(17 a + 17\right)\cdot 61^{2} + \left(30 a + 48\right)\cdot 61^{3} + \left(26 a + 19\right)\cdot 61^{4} + \left(6 a + 53\right)\cdot 61^{5} + \left(37 a + 35\right)\cdot 61^{6} +O\left(61^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 28 + 19\cdot 61 + 17\cdot 61^{2} + 40\cdot 61^{3} + 22\cdot 61^{4} + 61^{5} + 9\cdot 61^{6} +O\left(61^{ 7 }\right)$

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

Cycle notation

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

$(1,6)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.