Properties

Label 2.180.8t11.b.b
Dimension $2$
Group $Q_8:C_2$
Conductor $180$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Artin stem field: 8.0.4665600.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.20.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{15})\)

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 84\cdot 109 + 49\cdot 109^{2} + 42\cdot 109^{3} + 38\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 63\cdot 109 + 17\cdot 109^{2} + 87\cdot 109^{3} + 75\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 81\cdot 109 + 92\cdot 109^{2} + 9\cdot 109^{3} + 93\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 67 + 2\cdot 109 + 59\cdot 109^{2} + 90\cdot 109^{3} + 10\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 73 + 70\cdot 109 + 42\cdot 109^{2} + 104\cdot 109^{3} + 2\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 75 + 102\cdot 109 + 35\cdot 109^{2} + 67\cdot 109^{3} + 85\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 90 + 37\cdot 109 + 107\cdot 109^{2} + 36\cdot 109^{3} + 35\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 99 + 102\cdot 109 + 30\cdot 109^{2} + 106\cdot 109^{3} + 93\cdot 109^{4} +O(109^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.