Artin representation 1.5_7_19.4t1.1c1

• Label: 1.5_7_19.4t1.1c1
• Dimension: 1
• Group: $C_4$
• Conductor: $ 5 \cdot 7 \cdot 19 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$665= 5 \cdot 7 \cdot 19 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 166 x^{2} - 166 x + 5611 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Odd
Corresponding Dirichlet character:	\(\chi_{665}(132,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8\cdot 31 + 24\cdot 31^{2} + 13\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 1 + 3\cdot 31 + 29\cdot 31^{2} + 3\cdot 31^{3} + 11\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 12 + 3\cdot 31 + 14\cdot 31^{2} + 27\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 19 + 16\cdot 31 + 25\cdot 31^{2} + 16\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

Cycle notation

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

$(1,2,4,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.