Artin representation 1.17_19.4t1.1c1

• Label: 1.17_19.4t1.1c1
• Dimension: 1
• Group: $C_4$
• Conductor: $ 17 \cdot 19 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$323= 17 \cdot 19 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 79 x^{2} + x + 1616 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Odd
Corresponding Dirichlet character:	\(\chi_{323}(132,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 6 + 9\cdot 47 + 42\cdot 47^{2} + 36\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 22 + 36\cdot 47 + 39\cdot 47^{2} + 40\cdot 47^{3} + 24\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 24 + 24\cdot 47 + 45\cdot 47^{2} + 30\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 43 + 23\cdot 47 + 13\cdot 47^{2} + 32\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 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.