Artin representation 3.2e8_5e2_7e2.6t8.5c1

• Label: 3.2e8_5e2_7e2.6t8.5c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{8} \cdot 5^{2} \cdot 7^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$313600= 2^{8} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{4} - 4 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:

$r_{ 1 }$	$=$	$ 26 + 107\cdot 157 + 23\cdot 157^{2} + 69\cdot 157^{3} + 102\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 32 + 38\cdot 157 + 90\cdot 157^{2} + 18\cdot 157^{3} + 113\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 122 + 140\cdot 157 + 76\cdot 157^{2} + 5\cdot 157^{3} + 73\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 134 + 27\cdot 157 + 123\cdot 157^{2} + 63\cdot 157^{3} + 25\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.