Artin representation 4.2e12_3e2_5e2.8t22.13c1

• Label: 4.2e12_3e2_5e2.8t22.13c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$921600= 2^{12} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 12 x^{6} - 12 x^{5} + 36 x^{4} + 72 x^{3} + 40 x^{2} + 12 x + 9 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 769 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 10 + 52\cdot 769 + 394\cdot 769^{2} + 194\cdot 769^{3} + 463\cdot 769^{4} + 354\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 123 + 529\cdot 769 + 649\cdot 769^{2} + 543\cdot 769^{3} + 625\cdot 769^{4} + 733\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 166 + 185\cdot 769 + 494\cdot 769^{2} + 494\cdot 769^{3} + 29\cdot 769^{4} + 16\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 267 + 425\cdot 769 + 318\cdot 769^{2} + 397\cdot 769^{3} + 690\cdot 769^{4} + 457\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 309 + 320\cdot 769 + 149\cdot 769^{2} + 590\cdot 769^{3} + 525\cdot 769^{4} + 312\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 327 + 636\cdot 769 + 344\cdot 769^{2} + 209\cdot 769^{3} + 692\cdot 769^{4} + 136\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 369 + 531\cdot 769 + 175\cdot 769^{2} + 402\cdot 769^{3} + 527\cdot 769^{4} + 760\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 736 + 395\cdot 769 + 549\cdot 769^{2} + 243\cdot 769^{3} + 290\cdot 769^{4} + 303\cdot 769^{5} +O\left(769^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.