Properties

Label 2.784.8t6.a.b
Dimension $2$
Group $D_{8}$
Conductor $784$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(784\)\(\medspace = 2^{4} \cdot 7^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.843308032.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $D_4$
Projective stem field: Galois closure of 4.0.1372.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 7x^{6} - 14x^{4} + 14x^{3} + 7x^{2} - 18x + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 12 + 74\cdot 197 + 187\cdot 197^{2} + 73\cdot 197^{3} + 4\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 + 104\cdot 197 + 182\cdot 197^{2} + 107\cdot 197^{3} + 6\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 51 + 31\cdot 197 + 41\cdot 197^{2} + 136\cdot 197^{3} + 120\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 56 + 115\cdot 197 + 33\cdot 197^{2} + 54\cdot 197^{3} + 65\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 59 + 152\cdot 197 + 74\cdot 197^{2} + 49\cdot 197^{3} + 85\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 106 + 184\cdot 197 + 179\cdot 197^{2} + 75\cdot 197^{3} + 65\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 122 + 68\cdot 197 + 16\cdot 197^{2} + 168\cdot 197^{3} + 177\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 159 + 57\cdot 197 + 72\cdot 197^{2} + 122\cdot 197^{3} + 65\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.