Artin representation 4.3e2_7e2_13e2.8t26.3c1

• Label: 4.3e2_7e2_13e2.8t26.3c1
• Dimension: 4
• Group: $(C_4^2 : C_2):C_2$
• Conductor: $ 3^{2} \cdot 7^{2} \cdot 13^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_4^2 : C_2):C_2$
Conductor:	$74529= 3^{2} \cdot 7^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + 5 x^{4} - 3 x^{3} + x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_4^2 : C_2):C_2$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 49 a + 5 + \left(40 a + 56\right)\cdot 101 + \left(95 a + 32\right)\cdot 101^{2} + \left(31 a + 15\right)\cdot 101^{3} + \left(81 a + 73\right)\cdot 101^{4} + \left(54 a + 1\right)\cdot 101^{5} + \left(38 a + 58\right)\cdot 101^{6} + \left(59 a + 17\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 16 + 2\cdot 101 + 79\cdot 101^{2} + 28\cdot 101^{3} + 43\cdot 101^{4} + 87\cdot 101^{5} + 24\cdot 101^{6} + 27\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 82 a + 87 + \left(7 a + 13\right)\cdot 101 + \left(64 a + 26\right)\cdot 101^{2} + \left(74 a + 53\right)\cdot 101^{3} + \left(91 a + 88\right)\cdot 101^{4} + \left(44 a + 36\right)\cdot 101^{5} + \left(26 a + 13\right)\cdot 101^{6} + \left(76 a + 97\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 19 a + 11 + \left(93 a + 64\right)\cdot 101 + \left(36 a + 72\right)\cdot 101^{2} + \left(26 a + 85\right)\cdot 101^{3} + \left(9 a + 77\right)\cdot 101^{4} + \left(56 a + 23\right)\cdot 101^{5} + \left(74 a + 74\right)\cdot 101^{6} + \left(24 a + 72\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 52 a + 100 + \left(60 a + 67\right)\cdot 101 + \left(5 a + 70\right)\cdot 101^{2} + \left(69 a + 47\right)\cdot 101^{3} + \left(19 a + 63\right)\cdot 101^{4} + \left(46 a + 38\right)\cdot 101^{5} + \left(62 a + 56\right)\cdot 101^{6} + \left(41 a + 14\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 10 a + 41 + \left(72 a + 11\right)\cdot 101 + \left(47 a + 19\right)\cdot 101^{2} + \left(77 a + 34\right)\cdot 101^{3} + \left(96 a + 19\right)\cdot 101^{4} + \left(78 a + 39\right)\cdot 101^{5} + \left(86 a + 32\right)\cdot 101^{6} + \left(60 a + 57\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 65 + 100\cdot 101 + 66\cdot 101^{2} + 44\cdot 101^{3} + 12\cdot 101^{4} + 19\cdot 101^{5} + 46\cdot 101^{6} + 4\cdot 101^{7} +O\left(101^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 91 a + 81 + \left(28 a + 87\right)\cdot 101 + \left(53 a + 36\right)\cdot 101^{2} + \left(23 a + 94\right)\cdot 101^{3} + \left(4 a + 25\right)\cdot 101^{4} + \left(22 a + 56\right)\cdot 101^{5} + \left(14 a + 98\right)\cdot 101^{6} + \left(40 a + 11\right)\cdot 101^{7} +O\left(101^{ 8 }\right)$

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

Cycle notation

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

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

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

$(2,7)(6,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.