# Properties

 Label 2.1400.8t6.b.b 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: 8.0.19208000000.2 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

 $f(x)$ $=$ $$x^{8} - 5 x^{6} + 45 x^{4} - 75 x^{2} + 50$$  .

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 7.

Roots:
 $r_{ 1 }$ $=$ $$24 + 78\cdot 113 + 107\cdot 113^{2} + 41\cdot 113^{3} + 70\cdot 113^{4} + 109\cdot 113^{5} + 31\cdot 113^{6} +O(113^{7})$$ $r_{ 2 }$ $=$ $$48 + 113 + 65\cdot 113^{2} + 9\cdot 113^{3} + 89\cdot 113^{4} + 72\cdot 113^{5} + 46\cdot 113^{6} +O(113^{7})$$ $r_{ 3 }$ $=$ $$54 + 5\cdot 113 + 98\cdot 113^{2} + 2\cdot 113^{3} + 35\cdot 113^{4} + 111\cdot 113^{5} + 47\cdot 113^{6} +O(113^{7})$$ $r_{ 4 }$ $=$ $$56 + 52\cdot 113 + 16\cdot 113^{2} + 90\cdot 113^{3} + 12\cdot 113^{4} + 52\cdot 113^{5} + 44\cdot 113^{6} +O(113^{7})$$ $r_{ 5 }$ $=$ $$57 + 60\cdot 113 + 96\cdot 113^{2} + 22\cdot 113^{3} + 100\cdot 113^{4} + 60\cdot 113^{5} + 68\cdot 113^{6} +O(113^{7})$$ $r_{ 6 }$ $=$ $$59 + 107\cdot 113 + 14\cdot 113^{2} + 110\cdot 113^{3} + 77\cdot 113^{4} + 113^{5} + 65\cdot 113^{6} +O(113^{7})$$ $r_{ 7 }$ $=$ $$65 + 111\cdot 113 + 47\cdot 113^{2} + 103\cdot 113^{3} + 23\cdot 113^{4} + 40\cdot 113^{5} + 66\cdot 113^{6} +O(113^{7})$$ $r_{ 8 }$ $=$ $$89 + 34\cdot 113 + 5\cdot 113^{2} + 71\cdot 113^{3} + 42\cdot 113^{4} + 3\cdot 113^{5} + 81\cdot 113^{6} +O(113^{7})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.