Properties

Label 2.1764.8t6.a
Dimension $2$
Group $D_{8}$
Conductor $1764$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.38423222208.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: Galois closure of 4.0.12348.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 13 + 205\cdot 233 + 123\cdot 233^{2} + 161\cdot 233^{3} + 158\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 36 + 167\cdot 233 + 231\cdot 233^{2} + 53\cdot 233^{3} + 94\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 83 + 16\cdot 233 + 61\cdot 233^{2} + 58\cdot 233^{3} + 80\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 115 + 89\cdot 233 + 174\cdot 233^{2} + 158\cdot 233^{3} + 211\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 146 + 193\cdot 233 + 110\cdot 233^{2} + 110\cdot 233^{3} + 185\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 154 + 101\cdot 233 + 8\cdot 233^{2} + 169\cdot 233^{3} + 19\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 183 + 73\cdot 233^{2} + 138\cdot 233^{3} + 58\cdot 233^{4} +O\left(233^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 205 + 157\cdot 233 + 148\cdot 233^{2} + 81\cdot 233^{3} + 123\cdot 233^{4} +O\left(233^{ 5 }\right)$

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

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

Character values on conjugacy classes

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