# Properties

 Label 2.1400.8t6.a.a Dimension $2$ Group $D_{8}$ Conductor $1400$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$1400$$$$\medspace = 2^{3} \cdot 5^{2} \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 8.0.19208000000.1 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Determinant: 1.56.2t1.b.a Projective image: $D_4$ Projective stem field: 4.0.9800.2

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} + 9 x^{6} - 27 x^{5} + 44 x^{4} - 94 x^{3} + 156 x^{2} - 88 x + 16$$  .

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$26 + 125\cdot 193 + 123\cdot 193^{2} + 186\cdot 193^{3} + 120\cdot 193^{4} + 38\cdot 193^{5} +O(193^{6})$$ $r_{ 2 }$ $=$ $$37 + 146\cdot 193 + 183\cdot 193^{2} + 85\cdot 193^{3} + 36\cdot 193^{4} + 61\cdot 193^{5} +O(193^{6})$$ $r_{ 3 }$ $=$ $$49 + 51\cdot 193 + 157\cdot 193^{2} + 15\cdot 193^{3} + 90\cdot 193^{4} + 6\cdot 193^{5} +O(193^{6})$$ $r_{ 4 }$ $=$ $$51 + 61\cdot 193 + 67\cdot 193^{2} + 133\cdot 193^{3} + 5\cdot 193^{4} +O(193^{6})$$ $r_{ 5 }$ $=$ $$59 + 82\cdot 193 + 180\cdot 193^{2} + 114\cdot 193^{3} + 92\cdot 193^{4} + 92\cdot 193^{5} +O(193^{6})$$ $r_{ 6 }$ $=$ $$86 + 29\cdot 193 + 154\cdot 193^{2} + 175\cdot 193^{3} + 77\cdot 193^{4} + 77\cdot 193^{5} +O(193^{6})$$ $r_{ 7 }$ $=$ $$130 + 136\cdot 193 + 88\cdot 193^{2} + 83\cdot 193^{3} + 57\cdot 193^{4} + 36\cdot 193^{5} +O(193^{6})$$ $r_{ 8 }$ $=$ $$142 + 139\cdot 193 + 9\cdot 193^{2} + 169\cdot 193^{3} + 97\cdot 193^{4} + 73\cdot 193^{5} +O(193^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.