Properties

Label 2.1152.8t11.b
Dimension $2$
Group $Q_8:C_2$
Conductor $1152$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(1152\)\(\medspace = 2^{7} \cdot 3^{2}\)
Artin number field: Galois closure of 8.0.191102976.3
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{6})\)

Galois action

Roots of defining polynomial

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 values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$
$2$ $2$ $(1,7)(2,8)(3,5)(4,6)$ $0$ $0$
$2$ $2$ $(2,7)(4,5)$ $0$ $0$
$2$ $2$ $(1,4)(2,6)(3,7)(5,8)$ $0$ $0$
$1$ $4$ $(1,3,8,6)(2,4,7,5)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$1$ $4$ $(1,6,8,3)(2,5,7,4)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$2$ $4$ $(1,4,8,5)(2,3,7,6)$ $0$ $0$
$2$ $4$ $(1,3,8,6)(2,5,7,4)$ $0$ $0$
$2$ $4$ $(1,7,8,2)(3,5,6,4)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.