Properties

Label 2.3072.8t6.b.a
Dimension $2$
Group $D_{8}$
Conductor $3072$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.57982058496.8
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.6144.1

Defining polynomial

$f(x)$$=$\(x^{8} + 24 x^{4} + 32 x^{2} + 24\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 34 + 219\cdot 283 + 78\cdot 283^{2} + 8\cdot 283^{3} + 118\cdot 283^{4} + 108\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 64 + 206\cdot 283 + 105\cdot 283^{2} + 262\cdot 283^{3} + 131\cdot 283^{4} + 197\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 113 + 270\cdot 283 + 98\cdot 283^{2} + 224\cdot 283^{3} + 231\cdot 283^{4} + 123\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 137 + 279\cdot 283 + 214\cdot 283^{2} + 66\cdot 283^{3} + 191\cdot 283^{4} + 210\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 146 + 3\cdot 283 + 68\cdot 283^{2} + 216\cdot 283^{3} + 91\cdot 283^{4} + 72\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 170 + 12\cdot 283 + 184\cdot 283^{2} + 58\cdot 283^{3} + 51\cdot 283^{4} + 159\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 219 + 76\cdot 283 + 177\cdot 283^{2} + 20\cdot 283^{3} + 151\cdot 283^{4} + 85\cdot 283^{5} +O(283^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 249 + 63\cdot 283 + 204\cdot 283^{2} + 274\cdot 283^{3} + 164\cdot 283^{4} + 174\cdot 283^{5} +O(283^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.