Artin representation 3.2e4_5e2_13e2.6t8.1c1

• Label: 3.2e4_5e2_13e2.6t8.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{4} \cdot 5^{2} \cdot 13^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$67600= 2^{4} \cdot 5^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{4} + 4 x^{2} - 2 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 $\Q_{ 157 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 4 + 101\cdot 157 + 82\cdot 157^{2} + 42\cdot 157^{3} + 130\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 40 + 90\cdot 157 + 111\cdot 157^{2} + 13\cdot 157^{3} + 79\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 44 + 118\cdot 157 + 36\cdot 157^{2} + 63\cdot 157^{3} + 93\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 69 + 4\cdot 157 + 83\cdot 157^{2} + 37\cdot 157^{3} + 11\cdot 157^{4} +O\left(157^{ 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.