Properties

Label 2.392.8t6.b
Dimension $2$
Group $D_{8}$
Conductor $392$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(392\)\(\medspace = 2^{3} \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.421654016.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: 4.0.2744.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 3 + 87\cdot 107 + 23\cdot 107^{2} + 3\cdot 107^{3} + 37\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 64\cdot 107 + 11\cdot 107^{2} + 30\cdot 107^{3} + 36\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 25 + 72\cdot 107 + 29\cdot 107^{2} + 23\cdot 107^{3} + 26\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 61 + 71\cdot 107 + 7\cdot 107^{2} + 14\cdot 107^{3} + 94\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 75 + 67\cdot 107 + 43\cdot 107^{2} + 81\cdot 107^{3} + 97\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 84 + 4\cdot 107 + 61\cdot 107^{2} + 21\cdot 107^{3} + 47\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 88 + 100\cdot 107 + 48\cdot 107^{2} + 59\cdot 107^{3} + 78\cdot 107^{4} +O(107^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 89 + 66\cdot 107 + 94\cdot 107^{2} + 87\cdot 107^{3} + 10\cdot 107^{4} +O(107^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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