Artin representation 4.3e2_7e2_19e4.8t26.1c1

• Label: 4.3e2_7e2_19e4.8t26.1c1
• Dimension: 4
• Group: $(C_4^2 : C_2):C_2$
• Conductor: $ 3^{2} \cdot 7^{2} \cdot 19^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_4^2 : C_2):C_2$
Conductor:	$57471561= 3^{2} \cdot 7^{2} \cdot 19^{4} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 3 x^{6} - 3 x^{5} - x^{4} + 6 x^{3} + 6 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 $\Q_{ 541 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 8 + 441\cdot 541 + 494\cdot 541^{2} + 25\cdot 541^{3} + 26\cdot 541^{4} + 283\cdot 541^{5} + 508\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 58 + 193\cdot 541 + 471\cdot 541^{2} + 507\cdot 541^{3} + 284\cdot 541^{4} + 104\cdot 541^{5} + 482\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 147 + 335\cdot 541 + 62\cdot 541^{2} + 115\cdot 541^{3} + 344\cdot 541^{4} + 413\cdot 541^{5} + 209\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 210 + 307\cdot 541 + 41\cdot 541^{2} + 541^{3} + 348\cdot 541^{4} + 266\cdot 541^{5} + 435\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 219 + 6\cdot 541 + 237\cdot 541^{2} + 236\cdot 541^{3} + 349\cdot 541^{4} + 8\cdot 541^{5} + 383\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 234 + 328\cdot 541 + 349\cdot 541^{2} + 166\cdot 541^{3} + 520\cdot 541^{4} + 56\cdot 541^{5} + 20\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 257 + 37\cdot 541 + 344\cdot 541^{2} + 277\cdot 541^{3} + 182\cdot 541^{4} + 130\cdot 541^{5} + 8\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 493 + 514\cdot 541 + 162\cdot 541^{2} + 292\cdot 541^{3} + 108\cdot 541^{4} + 359\cdot 541^{5} + 116\cdot 541^{6} +O\left(541^{ 7 }\right)$

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

Cycle notation

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

$(4,7)(6,8)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.