Properties

Label 2.164.8t6.a.a
Dimension $2$
Group $D_{8}$
Conductor $164$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(164\)\(\medspace = 2^{2} \cdot 41 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.17643776.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.164.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.656.1

Defining polynomial

$f(x)$$=$\(x^{8} + 3 x^{6} - 2 x^{3} - x^{2} + 2 x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 19 + 129\cdot 173 + 45\cdot 173^{2} + 42\cdot 173^{3} + 6\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 22 + 34\cdot 173 + 73\cdot 173^{2} + 24\cdot 173^{3} + 138\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 26 + 157\cdot 173 + 26\cdot 173^{2} + 149\cdot 173^{3} + 42\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 75 + 68\cdot 173 + 129\cdot 173^{2} + 126\cdot 173^{3} + 84\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 112 + 61\cdot 173 + 20\cdot 173^{2} + 87\cdot 173^{3} + 68\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 130 + 9\cdot 173 + 21\cdot 173^{2} + 75\cdot 173^{3} + 148\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 149 + 149\cdot 173 + 165\cdot 173^{2} + 131\cdot 173^{3} + 90\cdot 173^{4} +O(173^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 159 + 81\cdot 173 + 36\cdot 173^{2} + 55\cdot 173^{3} + 112\cdot 173^{4} +O(173^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,4)(2,8)(3,5)(6,7)$
$(1,7)(3,6)(5,8)$
$(1,8)(2,4)(3,6)(5,7)$
$(1,7,8,5)(2,6,4,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,4)(3,6)(5,7)$$-2$
$4$$2$$(1,4)(2,8)(3,5)(6,7)$$0$
$4$$2$$(1,7)(3,6)(5,8)$$0$
$2$$4$$(1,7,8,5)(2,6,4,3)$$0$
$2$$8$$(1,4,7,3,8,2,5,6)$$-\zeta_{8}^{3} + \zeta_{8}$
$2$$8$$(1,3,5,4,8,6,7,2)$$\zeta_{8}^{3} - \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.