Artin representation 4.3e2_5e2_11e2.8t26.3c1

• Label: 4.3e2_5e2_11e2.8t26.3c1
• Dimension: 4
• Group: $(C_4^2 : C_2):C_2$
• Conductor: $ 3^{2} \cdot 5^{2} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_4^2 : C_2):C_2$
Conductor:	$27225= 3^{2} \cdot 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - x^{6} + 6 x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 2 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 $\Q_{ 379 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 45 + 188\cdot 379 + 177\cdot 379^{2} + 174\cdot 379^{3} + 171\cdot 379^{4} + 276\cdot 379^{5} + 296\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 154 + 80\cdot 379 + 48\cdot 379^{2} + 155\cdot 379^{3} + 168\cdot 379^{4} + 296\cdot 379^{5} + 284\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 221 + 122\cdot 379 + 71\cdot 379^{2} + 172\cdot 379^{3} + 138\cdot 379^{4} + 191\cdot 379^{5} + 357\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 249 + 219\cdot 379 + 233\cdot 379^{2} + 192\cdot 379^{3} + 326\cdot 379^{4} + 19\cdot 379^{5} + 327\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 252 + 157\cdot 379 + 128\cdot 379^{2} + 270\cdot 379^{3} + 355\cdot 379^{4} + 321\cdot 379^{5} + 112\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 266 + 90\cdot 379 + 188\cdot 379^{2} + 35\cdot 379^{3} + 174\cdot 379^{4} + 124\cdot 379^{5} + 305\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 339 + 366\cdot 379 + 81\cdot 379^{2} + 256\cdot 379^{3} + 279\cdot 379^{4} + 372\cdot 379^{5} + 197\cdot 379^{6} +O\left(379^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 371 + 289\cdot 379 + 207\cdot 379^{2} + 259\cdot 379^{3} + 280\cdot 379^{4} + 291\cdot 379^{5} + 12\cdot 379^{6} +O\left(379^{ 7 }\right)$

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

Cycle notation

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

$(1,7)(6,8)$

$(2,3)(4,5)$

$(4,5)(6,8)$

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

$(4,8,5,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.