Properties

 Label 2.1280.8t6.c Dimension $2$ Group $D_{8}$ Conductor $1280$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$1280$$$$\medspace = 2^{8} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.0.2097152000.4 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: Galois closure of 4.0.1280.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 7.
Roots:
 $r_{ 1 }$ $=$ $$3 + 10\cdot 61 + 31\cdot 61^{2} + 16\cdot 61^{3} + 48\cdot 61^{4} + 57\cdot 61^{5} + 38\cdot 61^{6} +O(61^{7})$$ 3 + 10*61 + 31*61^2 + 16*61^3 + 48*61^4 + 57*61^5 + 38*61^6+O(61^7) $r_{ 2 }$ $=$ $$12 + 5\cdot 61 + 30\cdot 61^{2} + 30\cdot 61^{3} + 32\cdot 61^{4} + 46\cdot 61^{5} + 9\cdot 61^{6} +O(61^{7})$$ 12 + 5*61 + 30*61^2 + 30*61^3 + 32*61^4 + 46*61^5 + 9*61^6+O(61^7) $r_{ 3 }$ $=$ $$18 + 6\cdot 61 + 45\cdot 61^{2} + 54\cdot 61^{3} + 18\cdot 61^{4} + 7\cdot 61^{5} +O(61^{7})$$ 18 + 6*61 + 45*61^2 + 54*61^3 + 18*61^4 + 7*61^5+O(61^7) $r_{ 4 }$ $=$ $$25 + 21\cdot 61 + 37\cdot 61^{2} + 8\cdot 61^{4} + 48\cdot 61^{5} + 5\cdot 61^{6} +O(61^{7})$$ 25 + 21*61 + 37*61^2 + 8*61^4 + 48*61^5 + 5*61^6+O(61^7) $r_{ 5 }$ $=$ $$36 + 39\cdot 61 + 23\cdot 61^{2} + 60\cdot 61^{3} + 52\cdot 61^{4} + 12\cdot 61^{5} + 55\cdot 61^{6} +O(61^{7})$$ 36 + 39*61 + 23*61^2 + 60*61^3 + 52*61^4 + 12*61^5 + 55*61^6+O(61^7) $r_{ 6 }$ $=$ $$43 + 54\cdot 61 + 15\cdot 61^{2} + 6\cdot 61^{3} + 42\cdot 61^{4} + 53\cdot 61^{5} + 60\cdot 61^{6} +O(61^{7})$$ 43 + 54*61 + 15*61^2 + 6*61^3 + 42*61^4 + 53*61^5 + 60*61^6+O(61^7) $r_{ 7 }$ $=$ $$49 + 55\cdot 61 + 30\cdot 61^{2} + 30\cdot 61^{3} + 28\cdot 61^{4} + 14\cdot 61^{5} + 51\cdot 61^{6} +O(61^{7})$$ 49 + 55*61 + 30*61^2 + 30*61^3 + 28*61^4 + 14*61^5 + 51*61^6+O(61^7) $r_{ 8 }$ $=$ $$58 + 50\cdot 61 + 29\cdot 61^{2} + 44\cdot 61^{3} + 12\cdot 61^{4} + 3\cdot 61^{5} + 22\cdot 61^{6} +O(61^{7})$$ 58 + 50*61 + 29*61^2 + 44*61^3 + 12*61^4 + 3*61^5 + 22*61^6+O(61^7)

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

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

Character values on conjugacy classes

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