Properties

Label 2.278784.8t5.a.a
Dimension $2$
Group $Q_8$
Conductor $278784$
Root number $-1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(278784\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin field: Galois closure of 8.8.5416809268248576.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{33})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 132x^{6} + 2970x^{4} - 17424x^{2} + 17424 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 167 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 1 + 40\cdot 167 + 49\cdot 167^{2} + 38\cdot 167^{3} + 133\cdot 167^{4} + 124\cdot 167^{5} + 78\cdot 167^{6} + 37\cdot 167^{7} + 78\cdot 167^{8} + 18\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 14\cdot 167 + 96\cdot 167^{2} + 125\cdot 167^{3} + 72\cdot 167^{4} + 147\cdot 167^{5} + 97\cdot 167^{7} + 117\cdot 167^{8} + 94\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 53 + 95\cdot 167 + 74\cdot 167^{2} + 36\cdot 167^{3} + 92\cdot 167^{4} + 150\cdot 167^{5} + 134\cdot 167^{6} + 163\cdot 167^{7} + 68\cdot 167^{8} + 153\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 74 + 118\cdot 167 + 104\cdot 167^{2} + 16\cdot 167^{3} + 43\cdot 167^{4} + 94\cdot 167^{5} + 61\cdot 167^{6} + 139\cdot 167^{7} + 135\cdot 167^{8} + 3\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 93 + 48\cdot 167 + 62\cdot 167^{2} + 150\cdot 167^{3} + 123\cdot 167^{4} + 72\cdot 167^{5} + 105\cdot 167^{6} + 27\cdot 167^{7} + 31\cdot 167^{8} + 163\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 114 + 71\cdot 167 + 92\cdot 167^{2} + 130\cdot 167^{3} + 74\cdot 167^{4} + 16\cdot 167^{5} + 32\cdot 167^{6} + 3\cdot 167^{7} + 98\cdot 167^{8} + 13\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 153 + 152\cdot 167 + 70\cdot 167^{2} + 41\cdot 167^{3} + 94\cdot 167^{4} + 19\cdot 167^{5} + 166\cdot 167^{6} + 69\cdot 167^{7} + 49\cdot 167^{8} + 72\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 166 + 126\cdot 167 + 117\cdot 167^{2} + 128\cdot 167^{3} + 33\cdot 167^{4} + 42\cdot 167^{5} + 88\cdot 167^{6} + 129\cdot 167^{7} + 88\cdot 167^{8} + 148\cdot 167^{9} +O(167^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character valueComplex conjugation
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
$2$$4$$(1,5,8,4)(2,3,7,6)$$0$