Properties

Label 2.3920.8t11.c
Dimension $2$
Group $Q_8:C_2$
Conductor $3920$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(3920\)\(\medspace = 2^{4} \cdot 5 \cdot 7^{2}\)
Artin number field: Galois closure of 8.0.12047257600.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{35})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 2 + 70\cdot 109 + 98\cdot 109^{2} + 57\cdot 109^{3} + 46\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 104\cdot 109^{2} + 41\cdot 109^{3} + 49\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 28 + 56\cdot 109 + 99\cdot 109^{2} + 35\cdot 109^{3} + 41\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 33 + 77\cdot 109 + 75\cdot 109^{2} + 31\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 44 + 84\cdot 109 + 47\cdot 109^{2} + 108\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 53 + 45\cdot 109 + 19\cdot 109^{2} + 32\cdot 109^{3} + 70\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 75 + 91\cdot 109 + 24\cdot 109^{2} + 82\cdot 109^{3} + 80\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 90 + 10\cdot 109 + 75\cdot 109^{2} + 45\cdot 109^{3} + 20\cdot 109^{4} +O(109^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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