# Properties

 Label 2.3600.8t11.c.b Dimension $2$ Group $Q_8:C_2$ Conductor $3600$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $$3600$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Artin stem field: Galois closure of 8.0.2916000000.5 Galois orbit size: $2$ Smallest permutation container: $Q_8:C_2$ Parity: odd Determinant: 1.20.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\zeta_{12})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 15x^{6} + 90x^{4} - 225x^{2} + 225$$ x^8 - 15*x^6 + 90*x^4 - 225*x^2 + 225 .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$2 + 30\cdot 61 + 7\cdot 61^{2} + 56\cdot 61^{3} + 3\cdot 61^{4} + 33\cdot 61^{5} +O(61^{6})$$ 2 + 30*61 + 7*61^2 + 56*61^3 + 3*61^4 + 33*61^5+O(61^6) $r_{ 2 }$ $=$ $$18 + 59\cdot 61 + 46\cdot 61^{2} + 20\cdot 61^{3} + 34\cdot 61^{4} + 38\cdot 61^{5} +O(61^{6})$$ 18 + 59*61 + 46*61^2 + 20*61^3 + 34*61^4 + 38*61^5+O(61^6) $r_{ 3 }$ $=$ $$19 + 10\cdot 61 + 50\cdot 61^{2} + 58\cdot 61^{3} + 19\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})$$ 19 + 10*61 + 50*61^2 + 58*61^3 + 19*61^4 + 22*61^5+O(61^6) $r_{ 4 }$ $=$ $$27 + 47\cdot 61 + 42\cdot 61^{2} + 26\cdot 61^{3} + 22\cdot 61^{4} + 54\cdot 61^{5} +O(61^{6})$$ 27 + 47*61 + 42*61^2 + 26*61^3 + 22*61^4 + 54*61^5+O(61^6) $r_{ 5 }$ $=$ $$34 + 13\cdot 61 + 18\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} + 6\cdot 61^{5} +O(61^{6})$$ 34 + 13*61 + 18*61^2 + 34*61^3 + 38*61^4 + 6*61^5+O(61^6) $r_{ 6 }$ $=$ $$42 + 50\cdot 61 + 10\cdot 61^{2} + 2\cdot 61^{3} + 41\cdot 61^{4} + 38\cdot 61^{5} +O(61^{6})$$ 42 + 50*61 + 10*61^2 + 2*61^3 + 41*61^4 + 38*61^5+O(61^6) $r_{ 7 }$ $=$ $$43 + 61 + 14\cdot 61^{2} + 40\cdot 61^{3} + 26\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})$$ 43 + 61 + 14*61^2 + 40*61^3 + 26*61^4 + 22*61^5+O(61^6) $r_{ 8 }$ $=$ $$59 + 30\cdot 61 + 53\cdot 61^{2} + 4\cdot 61^{3} + 57\cdot 61^{4} + 27\cdot 61^{5} +O(61^{6})$$ 59 + 30*61 + 53*61^2 + 4*61^3 + 57*61^4 + 27*61^5+O(61^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.