Artin representation 4.2e12_13e2.8t15.1c1

• Label: 4.2e12_13e2.8t15.1c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 13^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$692224= 2^{12} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} - 8 x^{4} + 22 x^{2} - 9 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 433 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 55 + 112\cdot 433 + 372\cdot 433^{2} + 22\cdot 433^{3} + 43\cdot 433^{4} + 215\cdot 433^{5} + 350\cdot 433^{6} + 259\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 56 + 237\cdot 433 + 235\cdot 433^{2} + 317\cdot 433^{3} + 410\cdot 433^{4} + 286\cdot 433^{5} + 278\cdot 433^{6} + 122\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 66 + 414\cdot 433 + 369\cdot 433^{2} + 230\cdot 433^{3} + 260\cdot 433^{4} + 96\cdot 433^{5} + 87\cdot 433^{6} + 167\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 136 + 81\cdot 433 + 253\cdot 433^{2} + 285\cdot 433^{3} + 420\cdot 433^{4} + 211\cdot 433^{5} + 134\cdot 433^{6} + 368\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 297 + 351\cdot 433 + 179\cdot 433^{2} + 147\cdot 433^{3} + 12\cdot 433^{4} + 221\cdot 433^{5} + 298\cdot 433^{6} + 64\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 367 + 18\cdot 433 + 63\cdot 433^{2} + 202\cdot 433^{3} + 172\cdot 433^{4} + 336\cdot 433^{5} + 345\cdot 433^{6} + 265\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 377 + 195\cdot 433 + 197\cdot 433^{2} + 115\cdot 433^{3} + 22\cdot 433^{4} + 146\cdot 433^{5} + 154\cdot 433^{6} + 310\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 378 + 320\cdot 433 + 60\cdot 433^{2} + 410\cdot 433^{3} + 389\cdot 433^{4} + 217\cdot 433^{5} + 82\cdot 433^{6} + 173\cdot 433^{7} +O\left(433^{ 8 }\right)$

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

Cycle notation

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

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

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

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

$(2,7)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.