Artin representation 4.2e10_7e4.8t26.4c1

• Label: 4.2e10_7e4.8t26.4c1
• Dimension: 4
• Group: $(C_4^2 : C_2):C_2$
• Conductor: $ 2^{10} \cdot 7^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_4^2 : C_2):C_2$
Conductor:	$2458624= 2^{10} \cdot 7^{4} $
Artin number field:	Splitting field of $f= x^{8} + 2 x^{6} - 4 x^{5} - 4 x^{3} + 2 x^{2} + 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 $\Q_{ 449 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 33 + 131\cdot 449 + 253\cdot 449^{2} + 434\cdot 449^{3} + 243\cdot 449^{4} + 390\cdot 449^{5} + 306\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 162 + 344\cdot 449 + 210\cdot 449^{2} + 91\cdot 449^{3} + 211\cdot 449^{4} + 124\cdot 449^{5} + 417\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 202 + 442\cdot 449 + 318\cdot 449^{2} + 448\cdot 449^{3} + 236\cdot 449^{4} + 60\cdot 449^{5} + 37\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 291 + 54\cdot 449 + 54\cdot 449^{2} + 292\cdot 449^{3} + 79\cdot 449^{4} + 236\cdot 449^{5} + 237\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 352 + 341\cdot 449 + 45\cdot 449^{2} + 135\cdot 449^{3} + 217\cdot 449^{4} + 14\cdot 449^{5} + 136\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 381 + 65\cdot 449 + 287\cdot 449^{2} + 283\cdot 449^{3} + 198\cdot 449^{4} + 273\cdot 449^{5} + 229\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 395 + 40\cdot 449 + 439\cdot 449^{2} + 444\cdot 449^{3} + 344\cdot 449^{4} + 11\cdot 449^{5} + 263\cdot 449^{6} +O\left(449^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 429 + 374\cdot 449 + 186\cdot 449^{2} + 114\cdot 449^{3} + 263\cdot 449^{4} + 235\cdot 449^{5} + 168\cdot 449^{6} +O\left(449^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.