# Properties

 Label 2.735.8t6.c.b Dimension $2$ Group $D_{8}$ Conductor $735$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$735$$$$\medspace = 3 \cdot 5 \cdot 7^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.2.8338372875.1 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Determinant: 1.15.2t1.a.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.15435.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 3x^{7} + 7x^{6} - 21x^{4} + 63x^{3} - 77x^{2} + 60x - 5$$ x^8 - 3*x^7 + 7*x^6 - 21*x^4 + 63*x^3 - 77*x^2 + 60*x - 5 .

The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$9 + 44\cdot 167 + 100\cdot 167^{2} + 50\cdot 167^{3} + 38\cdot 167^{4} +O(167^{5})$$ 9 + 44*167 + 100*167^2 + 50*167^3 + 38*167^4+O(167^5) $r_{ 2 }$ $=$ $$34 + 52\cdot 167 + 13\cdot 167^{2} + 51\cdot 167^{3} + 129\cdot 167^{4} +O(167^{5})$$ 34 + 52*167 + 13*167^2 + 51*167^3 + 129*167^4+O(167^5) $r_{ 3 }$ $=$ $$53 + 151\cdot 167 + 38\cdot 167^{2} + 21\cdot 167^{3} + 65\cdot 167^{4} +O(167^{5})$$ 53 + 151*167 + 38*167^2 + 21*167^3 + 65*167^4+O(167^5) $r_{ 4 }$ $=$ $$56 + 52\cdot 167 + 123\cdot 167^{2} + 90\cdot 167^{3} + 156\cdot 167^{4} +O(167^{5})$$ 56 + 52*167 + 123*167^2 + 90*167^3 + 156*167^4+O(167^5) $r_{ 5 }$ $=$ $$101 + 64\cdot 167 + 147\cdot 167^{2} + 122\cdot 167^{3} + 99\cdot 167^{4} +O(167^{5})$$ 101 + 64*167 + 147*167^2 + 122*167^3 + 99*167^4+O(167^5) $r_{ 6 }$ $=$ $$131 + 141\cdot 167 + 7\cdot 167^{2} + 50\cdot 167^{3} + 94\cdot 167^{4} +O(167^{5})$$ 131 + 141*167 + 7*167^2 + 50*167^3 + 94*167^4+O(167^5) $r_{ 7 }$ $=$ $$137 + 9\cdot 167 + 32\cdot 167^{2} + 109\cdot 167^{3} + 78\cdot 167^{4} +O(167^{5})$$ 137 + 9*167 + 32*167^2 + 109*167^3 + 78*167^4+O(167^5) $r_{ 8 }$ $=$ $$150 + 151\cdot 167 + 37\cdot 167^{2} + 5\cdot 167^{3} + 6\cdot 167^{4} +O(167^{5})$$ 150 + 151*167 + 37*167^2 + 5*167^3 + 6*167^4+O(167^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.