# Properties

 Label 2.1476.8t6.a Dimension $2$ Group $D_{8}$ Conductor $1476$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$1476$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 41$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.0.1429145856.1 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: 4.0.656.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$21 + 93\cdot 197 + 77\cdot 197^{2} + 102\cdot 197^{3} + 196\cdot 197^{4} +O(197^{5})$$ $r_{ 2 }$ $=$ $$39 + 150\cdot 197 + 15\cdot 197^{2} + 79\cdot 197^{3} + 172\cdot 197^{4} +O(197^{5})$$ $r_{ 3 }$ $=$ $$61 + 31\cdot 197 + 175\cdot 197^{2} + 38\cdot 197^{3} + 183\cdot 197^{4} +O(197^{5})$$ $r_{ 4 }$ $=$ $$101 + 8\cdot 197 + 162\cdot 197^{2} + 139\cdot 197^{3} + 29\cdot 197^{4} +O(197^{5})$$ $r_{ 5 }$ $=$ $$104 + 28\cdot 197 + 20\cdot 197^{2} + 173\cdot 197^{3} + 103\cdot 197^{4} +O(197^{5})$$ $r_{ 6 }$ $=$ $$112 + 79\cdot 197 + 82\cdot 197^{2} + 181\cdot 197^{3} + 119\cdot 197^{4} +O(197^{5})$$ $r_{ 7 }$ $=$ $$161 + 15\cdot 197 + 72\cdot 197^{2} + 167\cdot 197^{3} + 47\cdot 197^{4} +O(197^{5})$$ $r_{ 8 }$ $=$ $$191 + 183\cdot 197 + 182\cdot 197^{2} + 102\cdot 197^{3} + 131\cdot 197^{4} +O(197^{5})$$

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

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

### Character values on conjugacy classes

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