Artin representation 1.2e4_3_7.4t1.2c2

• Label: 1.2e4_3_7.4t1.2c2
• Dimension: 1
• Group: $C_4$
• Conductor: $ 2^{4} \cdot 3 \cdot 7 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$336= 2^{4} \cdot 3 \cdot 7 $
Artin number field:	Splitting field of $f= x^{4} + 84 x^{2} + 882 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Odd
Corresponding Dirichlet character:	\(\chi_{336}(83,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 4 + 16\cdot 17 + 6\cdot 17^{2} + 14\cdot 17^{3} + 9\cdot 17^{4} + 16\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 6 + 8\cdot 17^{2} + 17^{3} + 9\cdot 17^{4} + 3\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 11 + 16\cdot 17 + 8\cdot 17^{2} + 15\cdot 17^{3} + 7\cdot 17^{4} + 13\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 13 + 10\cdot 17^{2} + 2\cdot 17^{3} + 7\cdot 17^{4} +O\left(17^{ 6 }\right)$

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

Cycle notation

$(1,3,4,2)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.