Artin representation 3.2e10_5e2_17e2.6t8.3c1

• Label: 3.2e10_5e2_17e2.6t8.3c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{10} \cdot 5^{2} \cdot 17^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$7398400= 2^{10} \cdot 5^{2} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{4} - 10 x^{2} - 8 x - 2 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 26 + 95\cdot 179 + 138\cdot 179^{2} + 152\cdot 179^{3} + 119\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 + 98\cdot 179 + 127\cdot 179^{2} + 163\cdot 179^{3} + 108\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 29 + 53\cdot 179 + 22\cdot 179^{2} + 63\cdot 179^{3} + 78\cdot 179^{4} +O\left(179^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 97 + 111\cdot 179 + 69\cdot 179^{2} + 157\cdot 179^{3} + 50\cdot 179^{4} +O\left(179^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation

$(1,2,3,4)$

$(1,2)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.