Artin representation 1.2e3_3_5.4t1.2c1

• Label: 1.2e3_3_5.4t1.2c1
• Dimension: 1
• Group: $C_4$
• Conductor: $ 2^{3} \cdot 3 \cdot 5 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$1$
Group:	$C_4$
Conductor:	$120= 2^{3} \cdot 3 \cdot 5 $
Artin number field:	Splitting field of $f= x^{4} + 30 x^{2} + 180 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4$
Parity:	Odd
Corresponding Dirichlet character:	\(\chi_{120}(83,\cdot)\)

Galois action

Roots of defining polynomial

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

Roots:

$r_{ 1 }$	$=$	$ 3 + 35\cdot 59 + 25\cdot 59^{2} + 13\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 16 + 32\cdot 59 + 35\cdot 59^{2} + 48\cdot 59^{3} + 31\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 43 + 26\cdot 59 + 23\cdot 59^{2} + 10\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 56 + 23\cdot 59 + 33\cdot 59^{2} + 45\cdot 59^{3} + 51\cdot 59^{4} +O\left(59^{ 5 }\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.