Properties

Label 2.768.8t11.c
Dimension $2$
Group $Q_8:C_2$
Conductor $768$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(768\)\(\medspace = 2^{8} \cdot 3 \)
Artin number field: Galois closure of 8.0.150994944.1
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:
$r_{ 1 }$ $=$ \( 14 + 55\cdot 73 + 69\cdot 73^{2} + 53\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 15 + 67\cdot 73 + 48\cdot 73^{2} + 44\cdot 73^{3} + 57\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 34\cdot 73 + 19\cdot 73^{2} + 40\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 35 + 46\cdot 73 + 67\cdot 73^{2} + 69\cdot 73^{3} + 37\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 38 + 26\cdot 73 + 5\cdot 73^{2} + 3\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 57 + 38\cdot 73 + 53\cdot 73^{2} + 32\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 58 + 5\cdot 73 + 24\cdot 73^{2} + 28\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 59 + 17\cdot 73 + 3\cdot 73^{2} + 19\cdot 73^{3} + 47\cdot 73^{4} +O(73^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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