Properties

Label 2.475.8t8.a
Dimension $2$
Group $QD_{16}$
Conductor $475$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$QD_{16}$
Conductor:\(475\)\(\medspace = 5^{2} \cdot 19 \)
Artin number field: Galois closure of 8.2.107171875.1
Galois orbit size: $2$
Smallest permutation container: $QD_{16}$
Parity: odd
Projective image: $D_4$
Projective field: Galois closure of 4.2.475.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 3 + 86\cdot 199 + 183\cdot 199^{2} + 102\cdot 199^{3} + 51\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 + 64\cdot 199 + 101\cdot 199^{2} + 44\cdot 199^{3} + 158\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 42 + 179\cdot 199 + 134\cdot 199^{2} + 41\cdot 199^{3} + 53\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 83 + 38\cdot 199 + 130\cdot 199^{2} + 97\cdot 199^{3} + 9\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 149 + 176\cdot 199 + 136\cdot 199^{2} + 11\cdot 199^{3} + 129\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 156 + 18\cdot 199 + 141\cdot 199^{2} + 103\cdot 199^{3} + 124\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 164 + 96\cdot 199 + 146\cdot 199^{2} + 185\cdot 199^{3} + 8\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 171 + 135\cdot 199 + 20\cdot 199^{2} + 9\cdot 199^{3} + 62\cdot 199^{4} +O(199^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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