Properties

Label 2.441.8t8.a.b
Dimension $2$
Group $QD_{16}$
Conductor $441$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $QD_{16}$
Conductor: \(441\)\(\medspace = 3^{2} \cdot 7^{2}\)
Artin stem field: 8.2.257298363.1
Galois orbit size: $2$
Smallest permutation container: $QD_{16}$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.189.1

Defining polynomial

$f(x)$$=$\(x^{8} - 2 x^{7} + x^{6} - 5 x^{5} + 7 x^{4} - 2 x^{3} + x^{2} - 5 x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 39\cdot 79 + 56\cdot 79^{2} + 40\cdot 79^{3} + 4\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 22\cdot 79 + 60\cdot 79^{2} + 46\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 13\cdot 79 + 3\cdot 79^{2} + 65\cdot 79^{3} + 28\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 14 + 25\cdot 79 + 49\cdot 79^{2} + 17\cdot 79^{3} +O(79^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 39 + 53\cdot 79 + 46\cdot 79^{2} + 48\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 49 + 71\cdot 79 + 70\cdot 79^{2} + 52\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 51 + 11\cdot 79 + 21\cdot 79^{2} + 34\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 56 + 8\cdot 79^{2} + 10\cdot 79^{3} + 72\cdot 79^{4} +O(79^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.