Properties

Label 2.192.8t11.b
Dimension $2$
Group $Q_8:C_2$
Conductor $192$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(192\)\(\medspace = 2^{6} \cdot 3 \)
Artin number field: Galois closure of 8.0.5308416.2
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-2}, \sqrt{-3})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 6 + 32\cdot 73 + 55\cdot 73^{2} + 61\cdot 73^{3} + 29\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 26 + 58\cdot 73 + 44\cdot 73^{3} + 46\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 29 + 41\cdot 73 + 3\cdot 73^{2} + 16\cdot 73^{3} + 38\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 36 + 50\cdot 73 + 42\cdot 73^{2} + 41\cdot 73^{3} + 59\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 37 + 22\cdot 73 + 30\cdot 73^{2} + 31\cdot 73^{3} + 13\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 44 + 31\cdot 73 + 69\cdot 73^{2} + 56\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 47 + 14\cdot 73 + 72\cdot 73^{2} + 28\cdot 73^{3} + 26\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 67 + 40\cdot 73 + 17\cdot 73^{2} + 11\cdot 73^{3} + 43\cdot 73^{4} +O(73^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$
$2$ $2$ $(1,2)(3,5)(4,6)(7,8)$ $0$ $0$
$2$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $0$ $0$
$2$ $2$ $(2,7)(3,6)$ $0$ $0$
$1$ $4$ $(1,4,8,5)(2,6,7,3)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$1$ $4$ $(1,5,8,4)(2,3,7,6)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$2$ $4$ $(1,6,8,3)(2,4,7,5)$ $0$ $0$
$2$ $4$ $(1,4,8,5)(2,3,7,6)$ $0$ $0$
$2$ $4$ $(1,2,8,7)(3,4,6,5)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.