Properties

Label 2.441.8t8.a
Dimension $2$
Group $QD_{16}$
Conductor $441$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$QD_{16}$
Conductor:\(441\)\(\medspace = 3^{2} \cdot 7^{2}\)
Artin number field: Galois closure of 8.2.257298363.1
Galois orbit size: $2$
Smallest permutation container: $QD_{16}$
Parity: odd
Projective image: D_4
Projective field: 4.0.189.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 5 + 39\cdot 79 + 56\cdot 79^{2} + 40\cdot 79^{3} + 4\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 22\cdot 79 + 60\cdot 79^{2} + 46\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 13\cdot 79 + 3\cdot 79^{2} + 65\cdot 79^{3} + 28\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 14 + 25\cdot 79 + 49\cdot 79^{2} + 17\cdot 79^{3} +O(79^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 39 + 53\cdot 79 + 46\cdot 79^{2} + 48\cdot 79^{3} + 12\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 49 + 71\cdot 79 + 70\cdot 79^{2} + 52\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 51 + 11\cdot 79 + 21\cdot 79^{2} + 34\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 56 + 8\cdot 79^{2} + 10\cdot 79^{3} + 72\cdot 79^{4} +O(79^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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