Artin representation 3.2e2_11e2_53e2.6t8.3c1

• Label: 3.2e2_11e2_53e2.6t8.3c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 11^{2} \cdot 53^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$1359556= 2^{2} \cdot 11^{2} \cdot 53^{2} $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + x^{2} - 7 x - 4 $ 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 $\Q_{ 397 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 134 + 329\cdot 397 + 323\cdot 397^{2} + 177\cdot 397^{3} + 215\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 170 + 101\cdot 397 + 106\cdot 397^{2} + 286\cdot 397^{3} + 320\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 196 + 260\cdot 397 + 54\cdot 397^{2} + 15\cdot 397^{3} + 265\cdot 397^{4} +O\left(397^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 295 + 102\cdot 397 + 309\cdot 397^{2} + 314\cdot 397^{3} + 389\cdot 397^{4} +O\left(397^{ 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.