Artin representation 3.2e4_3_7e2_17.4t5.1c1

• Label: 3.2e4_3_7e2_17.4t5.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{4} \cdot 3 \cdot 7^{2} \cdot 17 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$39984= 2^{4} \cdot 3 \cdot 7^{2} \cdot 17 $
Artin number field:	Splitting field of $f= x^{4} - 4 x^{2} - 14 x - 10 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd
Determinant:	1.3_17.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 52 + 145\cdot 229 + 198\cdot 229^{2} + 219\cdot 229^{3} + 5\cdot 229^{4} +O\left(229^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 63 + 183\cdot 229 + 158\cdot 229^{2} + 210\cdot 229^{3} + 223\cdot 229^{4} +O\left(229^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 170 + 198\cdot 229 + 82\cdot 229^{2} + 121\cdot 229^{3} + 153\cdot 229^{4} +O\left(229^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 173 + 159\cdot 229 + 17\cdot 229^{2} + 135\cdot 229^{3} + 74\cdot 229^{4} +O\left(229^{ 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.