Artin representation 2.2e3_7_23e2.4t3.1c1

• Label: 2.2e3_7_23e2.4t3.1c1
• Dimension: 2
• Group: $D_{4}$
• Conductor: $ 2^{3} \cdot 7 \cdot 23^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$2$
Group:	$D_{4}$
Conductor:	$29624= 2^{3} \cdot 7 \cdot 23^{2} $
Artin number field:	Splitting field of $f= x^{4} - 2 x^{3} - 11 x^{2} - 34 x - 217 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$D_{4}$
Parity:	Odd
Determinant:	1.2e3_7.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 39 + 15\cdot 79 + 43\cdot 79^{2} + 25\cdot 79^{3} + 32\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 50 + 41\cdot 79 + 4\cdot 79^{2} + 8\cdot 79^{3} + 49\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 74 + 52\cdot 79 + 16\cdot 79^{2} + 79^{3} + 78\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 76 + 47\cdot 79 + 14\cdot 79^{2} + 44\cdot 79^{3} + 77\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

Cycle notation

$(1,2)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 4 }$	Character value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,2)(3,4)$	$-2$
$2$	$2$	$(1,3)(2,4)$	$0$
$2$	$2$	$(1,2)$	$0$
$2$	$4$	$(1,4,2,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.