# Properties

 Label 2.1215.9t3.a Dimension $2$ Group $D_{9}$ Conductor $1215$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{9}$ Conductor: $$1215$$$$\medspace = 3^{5} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.1.242137805625.3 Galois orbit size: $3$ Smallest permutation container: $D_{9}$ Parity: odd Projective image: $D_9$ Projective field: Galois closure of 9.1.242137805625.3

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $$x^{3} + 3x + 42$$
Roots:
 $r_{ 1 }$ $=$ $$31 a^{2} + 41 a + 15 + \left(33 a^{2} + 25 a + 20\right)\cdot 47 + \left(44 a^{2} + 7 a + 42\right)\cdot 47^{2} + \left(5 a^{2} + 11 a + 11\right)\cdot 47^{3} + \left(25 a^{2} + 35 a + 3\right)\cdot 47^{4} +O(47^{5})$$ 31*a^2 + 41*a + 15 + (33*a^2 + 25*a + 20)*47 + (44*a^2 + 7*a + 42)*47^2 + (5*a^2 + 11*a + 11)*47^3 + (25*a^2 + 35*a + 3)*47^4+O(47^5) $r_{ 2 }$ $=$ $$14 a^{2} + 39 a + 28 + \left(36 a^{2} + 6 a + 25\right)\cdot 47 + \left(35 a^{2} + 24 a + 24\right)\cdot 47^{2} + \left(38 a^{2} + 9 a + 30\right)\cdot 47^{3} + \left(14 a^{2} + 12 a + 29\right)\cdot 47^{4} +O(47^{5})$$ 14*a^2 + 39*a + 28 + (36*a^2 + 6*a + 25)*47 + (35*a^2 + 24*a + 24)*47^2 + (38*a^2 + 9*a + 30)*47^3 + (14*a^2 + 12*a + 29)*47^4+O(47^5) $r_{ 3 }$ $=$ $$6 a^{2} + 43 a + 12 + \left(42 a^{2} + 12 a + 37\right)\cdot 47 + \left(3 a^{2} + 3 a + 7\right)\cdot 47^{2} + \left(20 a^{2} + 14 a + 40\right)\cdot 47^{3} + 40 a\cdot 47^{4} +O(47^{5})$$ 6*a^2 + 43*a + 12 + (42*a^2 + 12*a + 37)*47 + (3*a^2 + 3*a + 7)*47^2 + (20*a^2 + 14*a + 40)*47^3 + 40*a*47^4+O(47^5) $r_{ 4 }$ $=$ $$13 a^{2} + 40 a + 26 + \left(27 a^{2} + 14 a + 7\right)\cdot 47 + \left(24 a^{2} + 27 a + 2\right)\cdot 47^{2} + \left(2 a^{2} + 34 a + 5\right)\cdot 47^{3} + \left(33 a^{2} + 9 a + 19\right)\cdot 47^{4} +O(47^{5})$$ 13*a^2 + 40*a + 26 + (27*a^2 + 14*a + 7)*47 + (24*a^2 + 27*a + 2)*47^2 + (2*a^2 + 34*a + 5)*47^3 + (33*a^2 + 9*a + 19)*47^4+O(47^5) $r_{ 5 }$ $=$ $$15 a^{2} + 5 a + 30 + \left(26 a^{2} + 19 a + 5\right)\cdot 47 + \left(29 a^{2} + 21 a + 12\right)\cdot 47^{2} + \left(30 a^{2} + 44 a + 14\right)\cdot 47^{3} + \left(15 a^{2} + a + 31\right)\cdot 47^{4} +O(47^{5})$$ 15*a^2 + 5*a + 30 + (26*a^2 + 19*a + 5)*47 + (29*a^2 + 21*a + 12)*47^2 + (30*a^2 + 44*a + 14)*47^3 + (15*a^2 + a + 31)*47^4+O(47^5) $r_{ 6 }$ $=$ $$28 a^{2} + 11 a + 9 + \left(24 a^{2} + 19 a + 2\right)\cdot 47 + \left(18 a^{2} + 16 a + 37\right)\cdot 47^{2} + \left(24 a^{2} + 45 a + 1\right)\cdot 47^{3} + \left(13 a^{2} + 43 a + 27\right)\cdot 47^{4} +O(47^{5})$$ 28*a^2 + 11*a + 9 + (24*a^2 + 19*a + 2)*47 + (18*a^2 + 16*a + 37)*47^2 + (24*a^2 + 45*a + 1)*47^3 + (13*a^2 + 43*a + 27)*47^4+O(47^5) $r_{ 7 }$ $=$ $$2 a^{2} + 14 a + 4 + \left(24 a^{2} + 14 a + 1\right)\cdot 47 + \left(13 a^{2} + 15 a + 27\right)\cdot 47^{2} + \left(2 a^{2} + 26 a + 4\right)\cdot 47^{3} + \left(7 a^{2} + 46 a + 14\right)\cdot 47^{4} +O(47^{5})$$ 2*a^2 + 14*a + 4 + (24*a^2 + 14*a + 1)*47 + (13*a^2 + 15*a + 27)*47^2 + (2*a^2 + 26*a + 4)*47^3 + (7*a^2 + 46*a + 14)*47^4+O(47^5) $r_{ 8 }$ $=$ $$31 a^{2} + 30 a + 15 + \left(10 a^{2} + 43 a + 21\right)\cdot 47 + \left(22 a^{2} + 6 a + 44\right)\cdot 47^{2} + \left(18 a^{2} + 28 a + 36\right)\cdot 47^{3} + \left(42 a^{2} + 11 a + 37\right)\cdot 47^{4} +O(47^{5})$$ 31*a^2 + 30*a + 15 + (10*a^2 + 43*a + 21)*47 + (22*a^2 + 6*a + 44)*47^2 + (18*a^2 + 28*a + 36)*47^3 + (42*a^2 + 11*a + 37)*47^4+O(47^5) $r_{ 9 }$ $=$ $$a^{2} + 12 a + 2 + \left(10 a^{2} + 31 a + 20\right)\cdot 47 + \left(42 a^{2} + 18 a + 37\right)\cdot 47^{2} + \left(44 a^{2} + 21 a + 42\right)\cdot 47^{3} + \left(35 a^{2} + 33 a + 24\right)\cdot 47^{4} +O(47^{5})$$ a^2 + 12*a + 2 + (10*a^2 + 31*a + 20)*47 + (42*a^2 + 18*a + 37)*47^2 + (44*a^2 + 21*a + 42)*47^3 + (35*a^2 + 33*a + 24)*47^4+O(47^5)

### Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(1,7,2)(3,6,4)(5,9,8)$ $(1,4,9,7,3,8,2,6,5)$ $(1,8)(2,5)(3,4)(7,9)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character values $c1$ $c2$ $c3$ $1$ $1$ $()$ $2$ $2$ $2$ $9$ $2$ $(1,8)(2,5)(3,4)(7,9)$ $0$ $0$ $0$ $2$ $3$ $(1,7,2)(3,6,4)(5,9,8)$ $-1$ $-1$ $-1$ $2$ $9$ $(1,4,9,7,3,8,2,6,5)$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ $\zeta_{9}^{5} + \zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ $2$ $9$ $(1,9,3,2,5,4,7,8,6)$ $\zeta_{9}^{5} + \zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ $2$ $9$ $(1,3,5,7,6,9,2,4,8)$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ $\zeta_{9}^{5} + \zeta_{9}^{4}$
The blue line marks the conjugacy class containing complex conjugation.