Artin representation 3.2e6_3e2_5e2_7.4t5.2c1

• Label: 3.2e6_3e2_5e2_7.4t5.2c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$100800= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} + 10 x^{2} + 12 x - 6 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd
Determinant:	1.7.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 20 + 32\cdot 137 + 126\cdot 137^{2} + 100\cdot 137^{3} + 6\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 53 + 97\cdot 137 + 107\cdot 137^{3} + 22\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 78 + 103\cdot 137 + 94\cdot 137^{2} + 41\cdot 137^{3} + 63\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 125 + 40\cdot 137 + 52\cdot 137^{2} + 24\cdot 137^{3} + 44\cdot 137^{4} +O\left(137^{ 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.