Artin representation 1.5_37.4t1.4c1

• Label: 1.5_37.4t1.4c1
• Dimension: 1
• Group: $C_4$
• Conductor: $ 5 \cdot 37 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$185= 5 \cdot 37 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 42 x^{2} - 44 x + 641 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Odd
Corresponding Dirichlet character:	\(\chi_{185}(154,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 27 + 37\cdot 67 + 39\cdot 67^{2} + 53\cdot 67^{3} + 23\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 55 + 52\cdot 67 + 50\cdot 67^{2} + 66\cdot 67^{3} + 41\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 57 + 65\cdot 67 + 5\cdot 67^{2} + 4\cdot 67^{3} + 25\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 63 + 44\cdot 67 + 37\cdot 67^{2} + 9\cdot 67^{3} + 43\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

Cycle notation

$(1,4,2,3)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.