# Properties

 Label 2.2951.9t3.a.c Dimension $2$ Group $D_{9}$ Conductor $2951$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{9}$ Conductor: $$2951$$$$\medspace = 13 \cdot 227$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.75836247976801.1 Galois orbit size: $3$ Smallest permutation container: $D_{9}$ Parity: odd Determinant: 1.2951.2t1.a.a Projective image: $D_9$ Projective stem field: Galois closure of 9.1.75836247976801.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 4x^{8} + 9x^{7} - 37x^{6} + 187x^{5} - 499x^{4} + 843x^{3} - 734x^{2} + 546x - 13$$ x^9 - 4*x^8 + 9*x^7 - 37*x^6 + 187*x^5 - 499*x^4 + 843*x^3 - 734*x^2 + 546*x - 13 .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $$x^{3} + x + 35$$

Roots:
 $r_{ 1 }$ $=$ $$10 a^{2} + 37 a + 9 + \left(11 a^{2} + 5 a + 38\right)\cdot 41 + \left(16 a + 15\right)\cdot 41^{2} + \left(10 a^{2} + 19 a + 21\right)\cdot 41^{3} + \left(16 a^{2} + 26 a + 3\right)\cdot 41^{4} +O(41^{5})$$ 10*a^2 + 37*a + 9 + (11*a^2 + 5*a + 38)*41 + (16*a + 15)*41^2 + (10*a^2 + 19*a + 21)*41^3 + (16*a^2 + 26*a + 3)*41^4+O(41^5) $r_{ 2 }$ $=$ $$20 a^{2} + 38 a + 33 + \left(12 a^{2} + 30 a + 15\right)\cdot 41 + \left(34 a^{2} + 12 a + 39\right)\cdot 41^{2} + \left(18 a^{2} + 26 a + 22\right)\cdot 41^{3} + \left(30 a^{2} + 28 a + 11\right)\cdot 41^{4} +O(41^{5})$$ 20*a^2 + 38*a + 33 + (12*a^2 + 30*a + 15)*41 + (34*a^2 + 12*a + 39)*41^2 + (18*a^2 + 26*a + 22)*41^3 + (30*a^2 + 28*a + 11)*41^4+O(41^5) $r_{ 3 }$ $=$ $$15 a^{2} + 4 a + 3 + \left(10 a^{2} + 10 a + 37\right)\cdot 41 + \left(26 a^{2} + 32 a + 39\right)\cdot 41^{2} + \left(40 a^{2} + 30 a + 1\right)\cdot 41^{3} + \left(20 a^{2} + 16 a + 30\right)\cdot 41^{4} +O(41^{5})$$ 15*a^2 + 4*a + 3 + (10*a^2 + 10*a + 37)*41 + (26*a^2 + 32*a + 39)*41^2 + (40*a^2 + 30*a + 1)*41^3 + (20*a^2 + 16*a + 30)*41^4+O(41^5) $r_{ 4 }$ $=$ $$25 a^{2} + 29 a + 19 + \left(13 a^{2} + 25 a + 12\right)\cdot 41 + \left(14 a^{2} + 29 a + 25\right)\cdot 41^{2} + \left(9 a^{2} + 37 a + 34\right)\cdot 41^{3} + \left(4 a^{2} + 7 a + 22\right)\cdot 41^{4} +O(41^{5})$$ 25*a^2 + 29*a + 19 + (13*a^2 + 25*a + 12)*41 + (14*a^2 + 29*a + 25)*41^2 + (9*a^2 + 37*a + 34)*41^3 + (4*a^2 + 7*a + 22)*41^4+O(41^5) $r_{ 5 }$ $=$ $$4 a^{2} + 19 a + 36 + \left(14 a^{2} + 20 a + 16\right)\cdot 41 + \left(40 a^{2} + 23 a + 2\right)\cdot 41^{2} + \left(8 a^{2} + 13 a + 30\right)\cdot 41^{3} + \left(2 a^{2} + 6 a + 33\right)\cdot 41^{4} +O(41^{5})$$ 4*a^2 + 19*a + 36 + (14*a^2 + 20*a + 16)*41 + (40*a^2 + 23*a + 2)*41^2 + (8*a^2 + 13*a + 30)*41^3 + (2*a^2 + 6*a + 33)*41^4+O(41^5) $r_{ 6 }$ $=$ $$34 a^{2} + 39 a + 2 + \left(35 a^{2} + 3 a + 13\right)\cdot 41 + \left(35 a^{2} + 15 a + 5\right)\cdot 41^{2} + \left(36 a^{2} + 10 a + 13\right)\cdot 41^{3} + \left(34 a^{2} + 23 a + 39\right)\cdot 41^{4} +O(41^{5})$$ 34*a^2 + 39*a + 2 + (35*a^2 + 3*a + 13)*41 + (35*a^2 + 15*a + 5)*41^2 + (36*a^2 + 10*a + 13)*41^3 + (34*a^2 + 23*a + 39)*41^4+O(41^5) $r_{ 7 }$ $=$ $$33 a^{2} + 39 a + 15 + \left(35 a^{2} + 26 a + 40\right)\cdot 41 + \left(19 a^{2} + 34 a + 21\right)\cdot 41^{2} + \left(4 a^{2} + 40 a + 32\right)\cdot 41^{3} + \left(26 a^{2} + 19\right)\cdot 41^{4} +O(41^{5})$$ 33*a^2 + 39*a + 15 + (35*a^2 + 26*a + 40)*41 + (19*a^2 + 34*a + 21)*41^2 + (4*a^2 + 40*a + 32)*41^3 + (26*a^2 + 19)*41^4+O(41^5) $r_{ 8 }$ $=$ $$17 a^{2} + 25 a + 31 + \left(14 a^{2} + 30 a + 30\right)\cdot 41 + \left(7 a^{2} + 4 a + 7\right)\cdot 41^{2} + \left(13 a^{2} + a + 19\right)\cdot 41^{3} + \left(8 a^{2} + 6 a + 10\right)\cdot 41^{4} +O(41^{5})$$ 17*a^2 + 25*a + 31 + (14*a^2 + 30*a + 30)*41 + (7*a^2 + 4*a + 7)*41^2 + (13*a^2 + a + 19)*41^3 + (8*a^2 + 6*a + 10)*41^4+O(41^5) $r_{ 9 }$ $=$ $$6 a^{2} + 16 a + 20 + \left(16 a^{2} + 9 a\right)\cdot 41 + \left(26 a^{2} + 36 a + 6\right)\cdot 41^{2} + \left(21 a^{2} + 24 a + 29\right)\cdot 41^{3} + \left(20 a^{2} + 6 a + 33\right)\cdot 41^{4} +O(41^{5})$$ 6*a^2 + 16*a + 20 + (16*a^2 + 9*a)*41 + (26*a^2 + 36*a + 6)*41^2 + (21*a^2 + 24*a + 29)*41^3 + (20*a^2 + 6*a + 33)*41^4+O(41^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $2$ $9$ $2$ $(1,2)(3,6)(4,5)(8,9)$ $0$ $2$ $3$ $(1,4,9)(2,8,5)(3,7,6)$ $-1$ $2$ $9$ $(1,5,7,4,2,6,9,8,3)$ $-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ $2$ $9$ $(1,7,2,9,3,5,4,6,8)$ $-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ $2$ $9$ $(1,2,3,4,8,7,9,5,6)$ $\zeta_{9}^{5} + \zeta_{9}^{4}$

The blue line marks the conjugacy class containing complex conjugation.