Artin representation 3.2e8_5e2_7.6t11.1c1

• Label: 3.2e8_5e2_7.6t11.1c1
• Dimension: 3
• Group: $S_4\times C_2$
• Conductor: $ 2^{8} \cdot 5^{2} \cdot 7 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4\times C_2$
Conductor:	$44800= 2^{8} \cdot 5^{2} \cdot 7 $
Artin number field:	Splitting field of $f= x^{6} + 7 x^{4} + 11 x^{2} - 5 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4\times C_2$
Parity:	Odd
Determinant:	1.7.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 40 a + 48 + \left(47 a + 50\right)\cdot 59 + \left(38 a + 34\right)\cdot 59^{2} + \left(54 a + 25\right)\cdot 59^{3} + \left(20 a + 21\right)\cdot 59^{4} + \left(39 a + 44\right)\cdot 59^{5} + \left(35 a + 27\right)\cdot 59^{6} + \left(25 a + 34\right)\cdot 59^{7} + \left(7 a + 31\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 2 }$	$=$	$ 19 a + 29 + \left(11 a + 58\right)\cdot 59 + \left(20 a + 25\right)\cdot 59^{2} + \left(4 a + 41\right)\cdot 59^{3} + \left(38 a + 46\right)\cdot 59^{4} + \left(19 a + 3\right)\cdot 59^{5} + \left(23 a + 24\right)\cdot 59^{6} + \left(33 a + 24\right)\cdot 59^{7} + \left(51 a + 13\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 3 }$	$=$	$ 11 + 28\cdot 59 + 11\cdot 59^{2} + 23\cdot 59^{3} + 34\cdot 59^{4} + 11\cdot 59^{5} + 27\cdot 59^{6} + 7\cdot 59^{7} + 43\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 4 }$	$=$	$ 19 a + 11 + \left(11 a + 8\right)\cdot 59 + \left(20 a + 24\right)\cdot 59^{2} + \left(4 a + 33\right)\cdot 59^{3} + \left(38 a + 37\right)\cdot 59^{4} + \left(19 a + 14\right)\cdot 59^{5} + \left(23 a + 31\right)\cdot 59^{6} + \left(33 a + 24\right)\cdot 59^{7} + \left(51 a + 27\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 5 }$	$=$	$ 40 a + 30 + 47 a\cdot 59 + \left(38 a + 33\right)\cdot 59^{2} + \left(54 a + 17\right)\cdot 59^{3} + \left(20 a + 12\right)\cdot 59^{4} + \left(39 a + 55\right)\cdot 59^{5} + \left(35 a + 34\right)\cdot 59^{6} + \left(25 a + 34\right)\cdot 59^{7} + \left(7 a + 45\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$
$r_{ 6 }$	$=$	$ 48 + 30\cdot 59 + 47\cdot 59^{2} + 35\cdot 59^{3} + 24\cdot 59^{4} + 47\cdot 59^{5} + 31\cdot 59^{6} + 51\cdot 59^{7} + 15\cdot 59^{8} +O\left(59^{ 9 }\right)$

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

Cycle notation

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

$(2,5)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.