Artin representation 3.2e2_3e2_5e2_37e2.6t8.2c1

• Label: 3.2e2_3e2_5e2_37e2.6t8.2c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$1232100= 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 14 x - 26 $ 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 $\Q_{ 127 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 23 + 119\cdot 127 + 100\cdot 127^{2} + 34\cdot 127^{3} + 65\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 103\cdot 127 + 66\cdot 127^{2} + 101\cdot 127^{3} + 114\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 35 + 91\cdot 127 + 39\cdot 127^{2} + 83\cdot 127^{3} + 102\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 46 + 67\cdot 127 + 46\cdot 127^{2} + 34\cdot 127^{3} + 98\cdot 127^{4} +O\left(127^{ 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.