Artin representation 3.2e2_23e2_37e2.6t8.1c1

• Label: 3.2e2_23e2_37e2.6t8.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 23^{2} \cdot 37^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$2896804= 2^{2} \cdot 23^{2} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 10 x^{2} + 8 x + 18 $ 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 $\Q_{ 269 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 152 + 58\cdot 269 + 113\cdot 269^{2} + 138\cdot 269^{3} + 80\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 181 + 244\cdot 269 + 155\cdot 269^{2} + 263\cdot 269^{3} + 257\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 221 + 160\cdot 269 + 17\cdot 269^{2} + 243\cdot 269^{3} + 129\cdot 269^{4} +O\left(269^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 254 + 73\cdot 269 + 251\cdot 269^{2} + 161\cdot 269^{3} + 69\cdot 269^{4} +O\left(269^{ 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.