Properties

Label 2.896.8t6.a
Dimension $2$
Group $D_{8}$
Conductor $896$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(896\)\(\medspace = 2^{7} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.1258815488.2
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: 4.0.1568.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ \( 7 + 38\cdot 71 + 45\cdot 71^{2} + 52\cdot 71^{3} + 50\cdot 71^{4} + 30\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 27\cdot 71 + 58\cdot 71^{2} + 42\cdot 71^{3} + 46\cdot 71^{4} + 34\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 27 + 16\cdot 71 + 17\cdot 71^{2} + 67\cdot 71^{3} + 31\cdot 71^{4} + 68\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 33 + 67\cdot 71 + 6\cdot 71^{2} + 2\cdot 71^{3} + 5\cdot 71^{4} + 51\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 38 + 3\cdot 71 + 64\cdot 71^{2} + 68\cdot 71^{3} + 65\cdot 71^{4} + 19\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 44 + 54\cdot 71 + 53\cdot 71^{2} + 3\cdot 71^{3} + 39\cdot 71^{4} + 2\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 57 + 43\cdot 71 + 12\cdot 71^{2} + 28\cdot 71^{3} + 24\cdot 71^{4} + 36\cdot 71^{5} +O(71^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 64 + 32\cdot 71 + 25\cdot 71^{2} + 18\cdot 71^{3} + 20\cdot 71^{4} + 40\cdot 71^{5} +O(71^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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