Artin representation 1.3_61.4t1.1c1

• Label: 1.3_61.4t1.1c1
• Dimension: 1
• Group: $C_4$
• Conductor: $ 3 \cdot 61 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$183= 3 \cdot 61 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} - 53 x^{2} + 141 x + 117 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Even
Corresponding Dirichlet character:	\(\chi_{183}(50,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 11\cdot 13 + 6\cdot 13^{2} + 8\cdot 13^{3} + 2\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 2 + 11\cdot 13 + 8\cdot 13^{3} + 11\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 3 + 7\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 4\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 9 + 9\cdot 13 + 12\cdot 13^{2} + 13^{3} + 7\cdot 13^{4} +O\left(13^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation

$(1,2)(3,4)$

$(1,3,2,4)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 4 }$	Character value
$1$	$1$	$()$	$1$
$1$	$2$	$(1,2)(3,4)$	$-1$
$1$	$4$	$(1,3,2,4)$	$\zeta_{4}$
$1$	$4$	$(1,4,2,3)$	$-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.