# Properties

 Label 2.57600.8t5.f Dimension $2$ Group $Q_8$ Conductor $57600$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$57600$$$$\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin number field: Galois closure of 8.0.7644119040000.1 Galois orbit size: $1$ Smallest permutation container: $Q_8$ Parity: even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{3})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 10.
Roots:
 $r_{ 1 }$ $=$ $$1 + 17\cdot 23 + 22\cdot 23^{2} + 23^{3} + 21\cdot 23^{4} + 17\cdot 23^{5} + 2\cdot 23^{6} + 6\cdot 23^{7} + 12\cdot 23^{8} + 4\cdot 23^{9} +O(23^{10})$$ 1 + 17*23 + 22*23^2 + 23^3 + 21*23^4 + 17*23^5 + 2*23^6 + 6*23^7 + 12*23^8 + 4*23^9+O(23^10) $r_{ 2 }$ $=$ $$3 + 11\cdot 23 + 23^{2} + 18\cdot 23^{3} + 21\cdot 23^{4} + 18\cdot 23^{5} + 23^{6} + 2\cdot 23^{7} + 17\cdot 23^{8} + 7\cdot 23^{9} +O(23^{10})$$ 3 + 11*23 + 23^2 + 18*23^3 + 21*23^4 + 18*23^5 + 23^6 + 2*23^7 + 17*23^8 + 7*23^9+O(23^10) $r_{ 3 }$ $=$ $$4 + 15\cdot 23 + 17\cdot 23^{2} + 18\cdot 23^{3} + 19\cdot 23^{4} + 18\cdot 23^{5} + 16\cdot 23^{6} + 8\cdot 23^{7} + 4\cdot 23^{8} + 18\cdot 23^{9} +O(23^{10})$$ 4 + 15*23 + 17*23^2 + 18*23^3 + 19*23^4 + 18*23^5 + 16*23^6 + 8*23^7 + 4*23^8 + 18*23^9+O(23^10) $r_{ 4 }$ $=$ $$11 + 22\cdot 23 + 2\cdot 23^{2} + 3\cdot 23^{3} + 22\cdot 23^{4} + 6\cdot 23^{5} + 7\cdot 23^{6} + 14\cdot 23^{7} + 15\cdot 23^{8} + 11\cdot 23^{9} +O(23^{10})$$ 11 + 22*23 + 2*23^2 + 3*23^3 + 22*23^4 + 6*23^5 + 7*23^6 + 14*23^7 + 15*23^8 + 11*23^9+O(23^10) $r_{ 5 }$ $=$ $$12 + 20\cdot 23^{2} + 19\cdot 23^{3} + 16\cdot 23^{5} + 15\cdot 23^{6} + 8\cdot 23^{7} + 7\cdot 23^{8} + 11\cdot 23^{9} +O(23^{10})$$ 12 + 20*23^2 + 19*23^3 + 16*23^5 + 15*23^6 + 8*23^7 + 7*23^8 + 11*23^9+O(23^10) $r_{ 6 }$ $=$ $$19 + 7\cdot 23 + 5\cdot 23^{2} + 4\cdot 23^{3} + 3\cdot 23^{4} + 4\cdot 23^{5} + 6\cdot 23^{6} + 14\cdot 23^{7} + 18\cdot 23^{8} + 4\cdot 23^{9} +O(23^{10})$$ 19 + 7*23 + 5*23^2 + 4*23^3 + 3*23^4 + 4*23^5 + 6*23^6 + 14*23^7 + 18*23^8 + 4*23^9+O(23^10) $r_{ 7 }$ $=$ $$20 + 11\cdot 23 + 21\cdot 23^{2} + 4\cdot 23^{3} + 23^{4} + 4\cdot 23^{5} + 21\cdot 23^{6} + 20\cdot 23^{7} + 5\cdot 23^{8} + 15\cdot 23^{9} +O(23^{10})$$ 20 + 11*23 + 21*23^2 + 4*23^3 + 23^4 + 4*23^5 + 21*23^6 + 20*23^7 + 5*23^8 + 15*23^9+O(23^10) $r_{ 8 }$ $=$ $$22 + 5\cdot 23 + 21\cdot 23^{3} + 23^{4} + 5\cdot 23^{5} + 20\cdot 23^{6} + 16\cdot 23^{7} + 10\cdot 23^{8} + 18\cdot 23^{9} +O(23^{10})$$ 22 + 5*23 + 21*23^3 + 23^4 + 5*23^5 + 20*23^6 + 16*23^7 + 10*23^8 + 18*23^9+O(23^10)

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

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

### Character values on conjugacy classes

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