Artin representation 3.2e2_3_11e2_13e2.4t5.1c1

• Label: 3.2e2_3_11e2_13e2.4t5.1c1
• Dimension: 3
• Group: $S_4$
• Conductor: $ 2^{2} \cdot 3 \cdot 11^{2} \cdot 13^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$3$
Group:	$S_4$
Conductor:	$245388= 2^{2} \cdot 3 \cdot 11^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{4} - 10 x^{2} - 26 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_4$
Parity:	Odd
Determinant:	1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 421 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 59 + 8\cdot 421 + 86\cdot 421^{2} + 221\cdot 421^{3} + 208\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 152 + 275\cdot 421 + 127\cdot 421^{2} + 24\cdot 421^{3} + 78\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 236 + 68\cdot 421 + 413\cdot 421^{2} + 402\cdot 421^{3} + 233\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 395 + 68\cdot 421 + 215\cdot 421^{2} + 193\cdot 421^{3} + 321\cdot 421^{4} +O\left(421^{ 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.