Artin representation 4.2e6_3e4_5e2.8t40.5c1

• Label: 4.2e6_3e4_5e2.8t40.5c1
• Dimension: 4
• Group: $Q_8:S_4$
• Conductor: $ 2^{6} \cdot 3^{4} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Q_8:S_4$
Conductor:	$129600= 2^{6} \cdot 3^{4} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 12 x^{6} - 22 x^{5} + 37 x^{4} - 42 x^{3} + 40 x^{2} - 22 x + 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Q_8:S_4$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 24.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 72 a + 13 + \left(31 a + 20\right)\cdot 97 + \left(64 a + 32\right)\cdot 97^{2} + \left(16 a + 72\right)\cdot 97^{3} + \left(59 a + 75\right)\cdot 97^{4} + \left(28 a + 63\right)\cdot 97^{5} + \left(82 a + 21\right)\cdot 97^{6} + \left(87 a + 94\right)\cdot 97^{7} + \left(90 a + 46\right)\cdot 97^{8} + \left(12 a + 87\right)\cdot 97^{9} + \left(10 a + 49\right)\cdot 97^{10} + \left(51 a + 76\right)\cdot 97^{11} + \left(38 a + 54\right)\cdot 97^{12} + \left(10 a + 62\right)\cdot 97^{13} + \left(16 a + 45\right)\cdot 97^{14} + \left(21 a + 94\right)\cdot 97^{15} + \left(54 a + 31\right)\cdot 97^{16} + \left(44 a + 53\right)\cdot 97^{17} + \left(48 a + 46\right)\cdot 97^{18} + \left(27 a + 10\right)\cdot 97^{19} + \left(33 a + 94\right)\cdot 97^{20} + \left(13 a + 9\right)\cdot 97^{21} + \left(47 a + 80\right)\cdot 97^{22} + \left(88 a + 27\right)\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 2 }$	$=$	$ 33 a + 43 + \left(35 a + 78\right)\cdot 97 + \left(5 a + 8\right)\cdot 97^{2} + \left(2 a + 7\right)\cdot 97^{3} + \left(71 a + 32\right)\cdot 97^{4} + \left(85 a + 64\right)\cdot 97^{5} + \left(32 a + 85\right)\cdot 97^{6} + \left(69 a + 70\right)\cdot 97^{7} + \left(45 a + 25\right)\cdot 97^{8} + \left(29 a + 3\right)\cdot 97^{9} + \left(6 a + 90\right)\cdot 97^{10} + \left(13 a + 3\right)\cdot 97^{11} + \left(16 a + 14\right)\cdot 97^{12} + \left(58 a + 58\right)\cdot 97^{13} + \left(4 a + 46\right)\cdot 97^{14} + \left(16 a + 28\right)\cdot 97^{15} + \left(3 a + 37\right)\cdot 97^{16} + \left(30 a + 66\right)\cdot 97^{17} + \left(89 a + 73\right)\cdot 97^{18} + \left(90 a + 15\right)\cdot 97^{19} + \left(40 a + 40\right)\cdot 97^{20} + \left(82 a + 42\right)\cdot 97^{21} + \left(28 a + 63\right)\cdot 97^{22} + \left(33 a + 61\right)\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 3 }$	$=$	$ 33 a + 22 + \left(35 a + 16\right)\cdot 97 + \left(5 a + 21\right)\cdot 97^{2} + \left(2 a + 93\right)\cdot 97^{3} + \left(71 a + 92\right)\cdot 97^{4} + \left(85 a + 17\right)\cdot 97^{5} + \left(32 a + 64\right)\cdot 97^{6} + \left(69 a + 86\right)\cdot 97^{7} + \left(45 a + 94\right)\cdot 97^{8} + \left(29 a + 12\right)\cdot 97^{9} + \left(6 a + 30\right)\cdot 97^{10} + \left(13 a + 86\right)\cdot 97^{11} + \left(16 a + 79\right)\cdot 97^{12} + \left(58 a + 93\right)\cdot 97^{13} + \left(4 a + 6\right)\cdot 97^{14} + \left(16 a + 57\right)\cdot 97^{15} + \left(3 a + 72\right)\cdot 97^{16} + \left(30 a + 3\right)\cdot 97^{17} + \left(89 a + 61\right)\cdot 97^{18} + \left(90 a + 79\right)\cdot 97^{19} + \left(40 a + 9\right)\cdot 97^{20} + \left(82 a + 13\right)\cdot 97^{21} + \left(28 a + 87\right)\cdot 97^{22} + \left(33 a + 30\right)\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 4 }$	$=$	$ 7 + 47\cdot 97 + 60\cdot 97^{2} + 28\cdot 97^{3} + 43\cdot 97^{4} + 58\cdot 97^{5} + 31\cdot 97^{6} + 61\cdot 97^{7} + 71\cdot 97^{8} + 32\cdot 97^{9} + 82\cdot 97^{10} + 78\cdot 97^{11} + 21\cdot 97^{12} + 79\cdot 97^{13} + 23\cdot 97^{14} + 13\cdot 97^{15} + 53\cdot 97^{16} + 17\cdot 97^{17} + 24\cdot 97^{18} + 96\cdot 97^{19} + 84\cdot 97^{20} + 63\cdot 97^{21} + 28\cdot 97^{22} + 8\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 5 }$	$=$	$ 64 a + 76 + \left(61 a + 80\right)\cdot 97 + \left(91 a + 75\right)\cdot 97^{2} + \left(94 a + 3\right)\cdot 97^{3} + \left(25 a + 4\right)\cdot 97^{4} + \left(11 a + 79\right)\cdot 97^{5} + \left(64 a + 32\right)\cdot 97^{6} + \left(27 a + 10\right)\cdot 97^{7} + \left(51 a + 2\right)\cdot 97^{8} + \left(67 a + 84\right)\cdot 97^{9} + \left(90 a + 66\right)\cdot 97^{10} + \left(83 a + 10\right)\cdot 97^{11} + \left(80 a + 17\right)\cdot 97^{12} + \left(38 a + 3\right)\cdot 97^{13} + \left(92 a + 90\right)\cdot 97^{14} + \left(80 a + 39\right)\cdot 97^{15} + \left(93 a + 24\right)\cdot 97^{16} + \left(66 a + 93\right)\cdot 97^{17} + \left(7 a + 35\right)\cdot 97^{18} + \left(6 a + 17\right)\cdot 97^{19} + \left(56 a + 87\right)\cdot 97^{20} + \left(14 a + 83\right)\cdot 97^{21} + \left(68 a + 9\right)\cdot 97^{22} + \left(63 a + 66\right)\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 6 }$	$=$	$ 64 a + 55 + \left(61 a + 18\right)\cdot 97 + \left(91 a + 88\right)\cdot 97^{2} + \left(94 a + 89\right)\cdot 97^{3} + \left(25 a + 64\right)\cdot 97^{4} + \left(11 a + 32\right)\cdot 97^{5} + \left(64 a + 11\right)\cdot 97^{6} + \left(27 a + 26\right)\cdot 97^{7} + \left(51 a + 71\right)\cdot 97^{8} + \left(67 a + 93\right)\cdot 97^{9} + \left(90 a + 6\right)\cdot 97^{10} + \left(83 a + 93\right)\cdot 97^{11} + \left(80 a + 82\right)\cdot 97^{12} + \left(38 a + 38\right)\cdot 97^{13} + \left(92 a + 50\right)\cdot 97^{14} + \left(80 a + 68\right)\cdot 97^{15} + \left(93 a + 59\right)\cdot 97^{16} + \left(66 a + 30\right)\cdot 97^{17} + \left(7 a + 23\right)\cdot 97^{18} + \left(6 a + 81\right)\cdot 97^{19} + \left(56 a + 56\right)\cdot 97^{20} + \left(14 a + 54\right)\cdot 97^{21} + \left(68 a + 33\right)\cdot 97^{22} + \left(63 a + 35\right)\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 7 }$	$=$	$ 91 + 49\cdot 97 + 36\cdot 97^{2} + 68\cdot 97^{3} + 53\cdot 97^{4} + 38\cdot 97^{5} + 65\cdot 97^{6} + 35\cdot 97^{7} + 25\cdot 97^{8} + 64\cdot 97^{9} + 14\cdot 97^{10} + 18\cdot 97^{11} + 75\cdot 97^{12} + 17\cdot 97^{13} + 73\cdot 97^{14} + 83\cdot 97^{15} + 43\cdot 97^{16} + 79\cdot 97^{17} + 72\cdot 97^{18} + 12\cdot 97^{20} + 33\cdot 97^{21} + 68\cdot 97^{22} + 88\cdot 97^{23} +O\left(97^{ 24 }\right)$
$r_{ 8 }$	$=$	$ 25 a + 85 + \left(65 a + 76\right)\cdot 97 + \left(32 a + 64\right)\cdot 97^{2} + \left(80 a + 24\right)\cdot 97^{3} + \left(37 a + 21\right)\cdot 97^{4} + \left(68 a + 33\right)\cdot 97^{5} + \left(14 a + 75\right)\cdot 97^{6} + \left(9 a + 2\right)\cdot 97^{7} + \left(6 a + 50\right)\cdot 97^{8} + \left(84 a + 9\right)\cdot 97^{9} + \left(86 a + 47\right)\cdot 97^{10} + \left(45 a + 20\right)\cdot 97^{11} + \left(58 a + 42\right)\cdot 97^{12} + \left(86 a + 34\right)\cdot 97^{13} + \left(80 a + 51\right)\cdot 97^{14} + \left(75 a + 2\right)\cdot 97^{15} + \left(42 a + 65\right)\cdot 97^{16} + \left(52 a + 43\right)\cdot 97^{17} + \left(48 a + 50\right)\cdot 97^{18} + \left(69 a + 86\right)\cdot 97^{19} + \left(63 a + 2\right)\cdot 97^{20} + \left(83 a + 87\right)\cdot 97^{21} + \left(49 a + 16\right)\cdot 97^{22} + \left(8 a + 69\right)\cdot 97^{23} +O\left(97^{ 24 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(2,6)(4,7)$

$(1,3,7,2,8,5,4,6)$

$(3,5)(4,7)$

$(1,4,3,8,7,5)(2,6)$

$(2,5,4)(3,7,6)$

$(1,6,3)(2,5,8)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character value
$1$	$1$	$()$	$4$
$1$	$2$	$(1,8)(2,6)(3,5)(4,7)$	$-4$
$6$	$2$	$(3,5)(4,7)$	$0$
$12$	$2$	$(1,4)(2,5)(3,6)(7,8)$	$0$
$24$	$2$	$(1,6)(2,8)(4,7)$	$0$
$32$	$3$	$(1,6,3)(2,5,8)$	$1$
$6$	$4$	$(1,2,8,6)(3,4,5,7)$	$0$
$6$	$4$	$(1,7,8,4)(2,3,6,5)$	$0$
$12$	$4$	$(1,2,8,6)$	$2$
$12$	$4$	$(1,2,8,6)(3,5)(4,7)$	$-2$
$32$	$6$	$(1,4,3,8,7,5)(2,6)$	$-1$
$24$	$8$	$(1,3,2,4,8,5,6,7)$	$0$
$24$	$8$	$(1,2,7,3,8,6,4,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.