Artin representation 1.5_89.4t1.2c2

• Label: 1.5_89.4t1.2c2
• Dimension: 1
• Group: $C_4$
• Conductor: $ 5 \cdot 89 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$445= 5 \cdot 89 $
Artin number field:	Splitting field of $f= x^{4} - x^{3} + 111 x^{2} - 111 x + 2531 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Odd
Corresponding Dirichlet character:	\(\chi_{445}(88,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 5 + 11\cdot 59 + 44\cdot 59^{2} + 5\cdot 59^{3} + 47\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 8 + 11\cdot 59 + 47\cdot 59^{2} + 37\cdot 59^{3} + 20\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 18 + 42\cdot 59 + 37\cdot 59^{2} + 7\cdot 59^{3} + 16\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 29 + 53\cdot 59 + 47\cdot 59^{2} + 7\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 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.