Properties

Label 2.712.8t6.b.a
Dimension $2$
Group $D_{8}$
Conductor $712$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(712\)\(\medspace = 2^{3} \cdot 89 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.2887553024.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.712.2t1.b.a
Projective image: $D_4$
Projective stem field: 4.0.5696.2

Defining polynomial

$f(x)$$=$\(x^{8} - 4 x^{6} + 13 x^{4} - 24 x^{3} + 20 x^{2} - 12 x + 9\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 30 + 6\cdot 179 + 42\cdot 179^{2} + 96\cdot 179^{3} + 64\cdot 179^{4} + 63\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 31 + 61\cdot 179 + 138\cdot 179^{2} + 101\cdot 179^{3} + 101\cdot 179^{4} + 121\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 113 + 27\cdot 179 + 17\cdot 179^{2} + 128\cdot 179^{3} + 62\cdot 179^{4} + 131\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 116 + 46\cdot 179 + 4\cdot 179^{2} + 111\cdot 179^{3} + 99\cdot 179^{4} + 31\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 126 + 96\cdot 179 + 49\cdot 179^{2} + 159\cdot 179^{3} + 171\cdot 179^{4} + 113\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 156 + 38\cdot 179 + 147\cdot 179^{2} + 32\cdot 179^{3} + 32\cdot 179^{4} + 66\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 159 + 103\cdot 179 + 145\cdot 179^{2} + 111\cdot 179^{3} + 177\cdot 179^{4} + 133\cdot 179^{5} +O(179^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 164 + 155\cdot 179 + 171\cdot 179^{2} + 153\cdot 179^{3} + 5\cdot 179^{4} + 54\cdot 179^{5} +O(179^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.