Properties

Label 2.3960.8t11.g
Dimension $2$
Group $Q_8:C_2$
Conductor $3960$
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Artin number field: Galois closure of 8.0.9032601600.2
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{-55})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 60 + 81\cdot 127 + 8\cdot 127^{2} + 92\cdot 127^{3} + 71\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 61 + 103\cdot 127 + 117\cdot 127^{2} + 100\cdot 127^{3} + 5\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 63 + 103\cdot 127 + 24\cdot 127^{2} + 11\cdot 127^{3} + 75\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 71 + 92\cdot 127 + 102\cdot 127^{2} + 49\cdot 127^{3} + 101\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 75 + 19\cdot 127 + 66\cdot 127^{2} + 51\cdot 127^{3} + 30\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 88 + 11\cdot 127 + 57\cdot 127^{2} + 80\cdot 127^{3} + 119\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 97 + 69\cdot 127 + 27\cdot 127^{2} + 21\cdot 127^{3} + 122\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 122 + 25\cdot 127 + 103\cdot 127^{2} + 100\cdot 127^{3} + 108\cdot 127^{4} +O(127^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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