Properties

Label 2.1152.8t11.b.b
Dimension $2$
Group $Q_8:C_2$
Conductor $1152$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(1152\)\(\medspace = 2^{7} \cdot 3^{2}\)
Artin stem field: 8.0.191102976.3
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{6})\)

Defining polynomial

$f(x)$$=$\(x^{8} - 6 x^{6} + 18 x^{4} - 36 x^{2} + 36\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 4\cdot 73 + 45\cdot 73^{2} + 4\cdot 73^{3} + 55\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 13 + 21\cdot 73 + 47\cdot 73^{2} + 60\cdot 73^{3} + 39\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 42\cdot 73 + 38\cdot 73^{2} + 48\cdot 73^{3} + 23\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 34 + 17\cdot 73 + 42\cdot 73^{2} + 52\cdot 73^{3} + 62\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 39 + 55\cdot 73 + 30\cdot 73^{2} + 20\cdot 73^{3} + 10\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 55 + 30\cdot 73 + 34\cdot 73^{2} + 24\cdot 73^{3} + 49\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 60 + 51\cdot 73 + 25\cdot 73^{2} + 12\cdot 73^{3} + 33\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 67 + 68\cdot 73 + 27\cdot 73^{2} + 68\cdot 73^{3} + 17\cdot 73^{4} +O(73^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.