Galois orbit of Artin representations 4.2e6_7e3_37e2.8t29.1

• Label: 4.2e6_7e3_37e2.8t29.1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 2^{6} \cdot 7^{3} \cdot 37^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$30052288= 2^{6} \cdot 7^{3} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{5} - 16 x^{4} + 24 x^{3} - 7 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 673 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 77 + 17\cdot 673 + 49\cdot 673^{2} + 483\cdot 673^{3} + 630\cdot 673^{4} + 512\cdot 673^{5} + 216\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 266 + 457\cdot 673 + 399\cdot 673^{2} + 412\cdot 673^{3} + 473\cdot 673^{4} + 207\cdot 673^{5} + 158\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 315 + 189\cdot 673 + 437\cdot 673^{2} + 473\cdot 673^{3} + 415\cdot 673^{4} + 389\cdot 673^{5} + 228\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 413 + 57\cdot 673 + 440\cdot 673^{2} + 124\cdot 673^{3} + 67\cdot 673^{4} + 185\cdot 673^{5} + 98\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 495 + 7\cdot 673 + 207\cdot 673^{2} + 40\cdot 673^{3} + 232\cdot 673^{4} + 10\cdot 673^{5} + 411\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 508 + 190\cdot 673 + 17\cdot 673^{2} + 410\cdot 673^{3} + 9\cdot 673^{4} + 615\cdot 673^{5} + 559\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 634 + 464\cdot 673 + 100\cdot 673^{2} + 566\cdot 673^{3} + 576\cdot 673^{4} + 496\cdot 673^{5} + 600\cdot 673^{6} +O\left(673^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 657 + 633\cdot 673 + 367\cdot 673^{2} + 181\cdot 673^{3} + 286\cdot 673^{4} + 274\cdot 673^{5} + 418\cdot 673^{6} +O\left(673^{ 7 }\right)$

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

Cycle notation

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

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

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

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

$(3,8)(4,7)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,5)(2,6)(3,8)(4,7)$	$-4$
$2$	$2$	$(1,5)(2,6)$	$0$
$2$	$2$	$(1,2)(3,4)(5,6)(7,8)$	$0$
$2$	$2$	$(1,2)(3,7)(4,8)(5,6)$	$0$
$4$	$2$	$(1,4)(2,8)(3,6)(5,7)$	$0$
$4$	$2$	$(1,5)(3,8)$	$0$
$4$	$2$	$(1,2)(5,6)$	$-2$
$4$	$2$	$(1,7)(2,8)(3,6)(4,5)$	$0$
$4$	$2$	$(1,2)(3,8)(4,7)(5,6)$	$2$
$4$	$4$	$(1,7,5,4)(2,8,6,3)$	$0$
$4$	$4$	$(1,4,5,7)(2,8,6,3)$	$0$
$4$	$4$	$(1,2,5,6)(3,7,8,4)$	$0$
$8$	$4$	$(1,2,5,6)(4,7)$	$0$
$8$	$4$	$(1,8,2,7)(3,6,4,5)$	$0$
$8$	$4$	$(1,8,2,4)(3,6,7,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.