Properties

Label 2.896.8t6.d.a
Dimension $2$
Group $D_{8}$
Conductor $896$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(896\)\(\medspace = 2^{7} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.1438646272.5
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.56.2t1.b.a
Projective image: $D_4$
Projective stem field: Galois closure of 4.0.1568.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} + 8x^{6} - 22x^{4} + 48x^{3} - 44x^{2} + 16x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 2 + 21\cdot 71 + 39\cdot 71^{2} + 50\cdot 71^{3} + 36\cdot 71^{4} + 70\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 47\cdot 71 + 53\cdot 71^{2} + 70\cdot 71^{3} + 9\cdot 71^{4} + 70\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 62\cdot 71 + 60\cdot 71^{2} + 2\cdot 71^{3} + 48\cdot 71^{4} + 46\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 29 + 63\cdot 71 + 3\cdot 71^{2} + 19\cdot 71^{3} + 48\cdot 71^{4} + 6\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 34 + 46\cdot 71^{2} + 68\cdot 71^{3} + 64\cdot 71^{4} + 63\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 36 + 10\cdot 71 + 45\cdot 71^{2} + 71^{3} + 47\cdot 71^{4} + 65\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 43 + 12\cdot 71 + 23\cdot 71^{2} + 24\cdot 71^{3} + 38\cdot 71^{4} + 57\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 46 + 66\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 61\cdot 71^{4} + 44\cdot 71^{5} +O(71^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.