Artin representation 4.3e4_5e2_7e4.8t26.2c1

• Label: 4.3e4_5e2_7e4.8t26.2c1
• Dimension: 4
• Group: $(C_4^2 : C_2):C_2$
• Conductor: $ 3^{4} \cdot 5^{2} \cdot 7^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_4^2 : C_2):C_2$
Conductor:	$4862025= 3^{4} \cdot 5^{2} \cdot 7^{4} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 6 x^{5} - 9 x^{3} + 6 x^{2} - 3 x + 3 $ 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 $\Q_{ 571 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 145 + 299\cdot 571 + 418\cdot 571^{2} + 19\cdot 571^{3} + 76\cdot 571^{4} + 507\cdot 571^{5} + 322\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 191 + 32\cdot 571 + 319\cdot 571^{2} + 262\cdot 571^{3} + 179\cdot 571^{4} + 104\cdot 571^{5} + 296\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 226 + 186\cdot 571 + 154\cdot 571^{2} + 28\cdot 571^{3} + 217\cdot 571^{4} + 562\cdot 571^{5} + 433\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 242 + 371\cdot 571 + 503\cdot 571^{2} + 196\cdot 571^{3} + 119\cdot 571^{4} + 478\cdot 571^{5} + 317\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 245 + 225\cdot 571 + 49\cdot 571^{2} + 105\cdot 571^{3} + 425\cdot 571^{4} + 555\cdot 571^{5} + 426\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 372 + 306\cdot 571 + 399\cdot 571^{2} + 384\cdot 571^{3} + 382\cdot 571^{4} + 356\cdot 571^{5} + 552\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 382 + 321\cdot 571 + 170\cdot 571^{2} + 67\cdot 571^{3} + 157\cdot 571^{4} + 324\cdot 571^{5} + 209\cdot 571^{6} +O\left(571^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 484 + 540\cdot 571 + 268\cdot 571^{2} + 77\cdot 571^{3} + 156\cdot 571^{4} + 537\cdot 571^{5} + 294\cdot 571^{6} +O\left(571^{ 7 }\right)$

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

Cycle notation

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

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

$(1,8)(2,5)$

$(3,6)(4,7)$

$(1,8)(4,7)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.