# Properties

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

# Related objects

## 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 stem field: 8.0.38423222208.1 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Determinant: 1.4.2t1.a.a Projective image: $D_4$ Projective stem field: 4.0.12348.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 3 x^{7} + 7 x^{6} - 21 x^{5} + 42 x^{4} - 42 x^{3} + 28 x^{2} - 24 x + 16$$  .

The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.

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

## 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

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,5)(4,6)(7,8)$ $-2$ $4$ $2$ $(1,7)(2,8)(4,6)$ $0$ $4$ $2$ $(1,3)(2,5)(4,8)(6,7)$ $0$ $2$ $4$ $(1,8,2,7)(3,6,5,4)$ $0$ $2$ $8$ $(1,6,8,5,2,4,7,3)$ $-\zeta_{8}^{3} + \zeta_{8}$ $2$ $8$ $(1,5,7,6,2,3,8,4)$ $\zeta_{8}^{3} - \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.