Properties

Label 2.48841.8t5.a.a
Dimension $2$
Group $Q_8$
Conductor $48841$
Root number $-1$
Indicator $-1$

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Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(48841\)\(\medspace = 13^{2} \cdot 17^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin field: Galois closure of 8.8.116507435287321.1
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 \(\Q(\sqrt{13}, \sqrt{17})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 3x^{7} - 67x^{6} - 16x^{5} + 863x^{4} + 1276x^{3} + 392x^{2} - 54x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 42 + 54\cdot 179 + 111\cdot 179^{2} + 6\cdot 179^{3} + 165\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 46 + 76\cdot 179 + 151\cdot 179^{2} + 6\cdot 179^{3} + 114\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 49 + 122\cdot 179 + 153\cdot 179^{2} + 19\cdot 179^{3} + 124\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 156\cdot 179 + 2\cdot 179^{2} + 169\cdot 179^{3} + 73\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 101 + 155\cdot 179 + 92\cdot 179^{2} + 101\cdot 179^{3} + 139\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 136 + 98\cdot 179 + 164\cdot 179^{2} + 124\cdot 179^{3} + 100\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 142 + 113\cdot 179 + 82\cdot 179^{2} + 47\cdot 179^{3} + 177\cdot 179^{4} +O(179^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 153 + 117\cdot 179 + 135\cdot 179^{2} + 60\cdot 179^{3} +O(179^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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