# Properties

 Label 1.33.10t1.b Dimension $1$ Group $C_{10}$ Conductor $33$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $$33$$$$\medspace = 3 \cdot 11$$ Artin number field: Galois closure of $$\Q(\zeta_{33})^+$$ Galois orbit size: $4$ Smallest permutation container: $C_{10}$ Parity: even Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{5} + x + 14$$
Roots:
 $r_{ 1 }$ $=$ $$14 a^{3} + 16 a^{2} + 14 a + 15 + \left(4 a^{4} + 11 a^{3} + 2 a^{2} + 5 a + 11\right)\cdot 17 + \left(9 a^{4} + 5 a^{3} + 3 a^{2} + 4 a + 1\right)\cdot 17^{2} + \left(14 a^{4} + 6 a^{3} + 4 a^{2} + 13 a + 14\right)\cdot 17^{3} + \left(15 a^{4} + 4 a^{3} + 11 a^{2} + 11 a + 1\right)\cdot 17^{4} +O(17^{5})$$ 14*a^3 + 16*a^2 + 14*a + 15 + (4*a^4 + 11*a^3 + 2*a^2 + 5*a + 11)*17 + (9*a^4 + 5*a^3 + 3*a^2 + 4*a + 1)*17^2 + (14*a^4 + 6*a^3 + 4*a^2 + 13*a + 14)*17^3 + (15*a^4 + 4*a^3 + 11*a^2 + 11*a + 1)*17^4+O(17^5) $r_{ 2 }$ $=$ $$2 a^{4} + a^{3} + 4 a^{2} + 16 a + 3 + \left(7 a^{4} + 11 a^{3} + 4 a^{2} + 11 a + 4\right)\cdot 17 + \left(10 a^{4} + 15 a^{3} + 11 a^{2} + 14 a + 16\right)\cdot 17^{2} + \left(2 a^{4} + 4 a^{3} + 9 a^{2} + 8 a + 7\right)\cdot 17^{3} + \left(6 a^{4} + 12 a^{3} + 9 a^{2} + 10 a + 14\right)\cdot 17^{4} +O(17^{5})$$ 2*a^4 + a^3 + 4*a^2 + 16*a + 3 + (7*a^4 + 11*a^3 + 4*a^2 + 11*a + 4)*17 + (10*a^4 + 15*a^3 + 11*a^2 + 14*a + 16)*17^2 + (2*a^4 + 4*a^3 + 9*a^2 + 8*a + 7)*17^3 + (6*a^4 + 12*a^3 + 9*a^2 + 10*a + 14)*17^4+O(17^5) $r_{ 3 }$ $=$ $$2 a^{4} + a^{3} + 12 a^{2} + 9 a + 14 + \left(4 a^{4} + 5 a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 17 + \left(14 a^{4} + 3 a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 17^{2} + \left(5 a^{4} + 3 a^{3} + 2 a + 5\right)\cdot 17^{3} + \left(9 a^{4} + 8 a^{3} + 9 a^{2} + 15 a + 1\right)\cdot 17^{4} +O(17^{5})$$ 2*a^4 + a^3 + 12*a^2 + 9*a + 14 + (4*a^4 + 5*a^3 + 11*a^2 + 8*a + 4)*17 + (14*a^4 + 3*a^3 + 11*a^2 + 5*a + 10)*17^2 + (5*a^4 + 3*a^3 + 2*a + 5)*17^3 + (9*a^4 + 8*a^3 + 9*a^2 + 15*a + 1)*17^4+O(17^5) $r_{ 4 }$ $=$ $$3 a^{4} + 13 a^{3} + 5 a^{2} + a + 14 + \left(8 a^{3} + 12 a^{2} + 8 a + 8\right)\cdot 17 + \left(15 a^{4} + 15 a^{3} + 15 a^{2} + 11 a + 16\right)\cdot 17^{2} + \left(6 a^{4} + 13 a^{3} + 14 a^{2} + 14 a + 7\right)\cdot 17^{3} + \left(11 a^{4} + 6 a^{3} + a^{2} + 10 a + 8\right)\cdot 17^{4} +O(17^{5})$$ 3*a^4 + 13*a^3 + 5*a^2 + a + 14 + (8*a^3 + 12*a^2 + 8*a + 8)*17 + (15*a^4 + 15*a^3 + 15*a^2 + 11*a + 16)*17^2 + (6*a^4 + 13*a^3 + 14*a^2 + 14*a + 7)*17^3 + (11*a^4 + 6*a^3 + a^2 + 10*a + 8)*17^4+O(17^5) $r_{ 5 }$ $=$ $$5 a^{4} + 15 a^{2} + 16 a + 13 + \left(6 a^{4} + 10 a^{3} + 6 a^{2} + 4 a + 16\right)\cdot 17 + \left(3 a^{4} + a^{3} + 13 a^{2} + 12 a + 4\right)\cdot 17^{2} + \left(10 a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 2\right)\cdot 17^{3} + \left(16 a^{4} + 11 a^{3} + 7 a^{2} + 14 a + 7\right)\cdot 17^{4} +O(17^{5})$$ 5*a^4 + 15*a^2 + 16*a + 13 + (6*a^4 + 10*a^3 + 6*a^2 + 4*a + 16)*17 + (3*a^4 + a^3 + 13*a^2 + 12*a + 4)*17^2 + (10*a^4 + 3*a^3 + 10*a^2 + 5*a + 2)*17^3 + (16*a^4 + 11*a^3 + 7*a^2 + 14*a + 7)*17^4+O(17^5) $r_{ 6 }$ $=$ $$5 a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 2 + \left(14 a^{4} + 14 a^{3} + 7 a^{2} + 14 a + 3\right)\cdot 17 + \left(a^{4} + 3 a^{3} + 15 a^{2} + 12 a + 16\right)\cdot 17^{2} + \left(5 a^{4} + 11 a^{3} + 3 a^{2} + 7 a + 9\right)\cdot 17^{3} + \left(12 a^{4} + 5 a^{3} + 6 a^{2} + 7 a + 12\right)\cdot 17^{4} +O(17^{5})$$ 5*a^4 + 6*a^3 + 3*a^2 + 7*a + 2 + (14*a^4 + 14*a^3 + 7*a^2 + 14*a + 3)*17 + (a^4 + 3*a^3 + 15*a^2 + 12*a + 16)*17^2 + (5*a^4 + 11*a^3 + 3*a^2 + 7*a + 9)*17^3 + (12*a^4 + 5*a^3 + 6*a^2 + 7*a + 12)*17^4+O(17^5) $r_{ 7 }$ $=$ $$7 a^{4} + 6 a^{2} + 13 a + 7 + \left(8 a^{4} + 5 a^{3} + 7 a^{2} + 10 a + 15\right)\cdot 17 + \left(14 a^{4} + 10 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 17^{2} + \left(4 a^{4} + 14 a^{3} + a^{2} + 6 a + 6\right)\cdot 17^{3} + \left(5 a^{4} + 4 a^{3} + 5 a^{2} + 10 a + 10\right)\cdot 17^{4} +O(17^{5})$$ 7*a^4 + 6*a^2 + 13*a + 7 + (8*a^4 + 5*a^3 + 7*a^2 + 10*a + 15)*17 + (14*a^4 + 10*a^3 + 5*a^2 + 7*a + 5)*17^2 + (4*a^4 + 14*a^3 + a^2 + 6*a + 6)*17^3 + (5*a^4 + 4*a^3 + 5*a^2 + 10*a + 10)*17^4+O(17^5) $r_{ 8 }$ $=$ $$7 a^{4} + 11 a^{3} + 3 a^{2} + 9 a + 1 + \left(10 a^{4} + 13 a^{3} + 3 a^{2} + 15 a + 3\right)\cdot 17 + \left(11 a^{4} + 9 a^{3} + 12 a^{2} + 8 a + 8\right)\cdot 17^{2} + \left(14 a^{4} + 12 a^{3} + 16 a^{2} + 13 a + 2\right)\cdot 17^{3} + \left(15 a^{4} + 4 a^{3} + 5 a^{2} + a + 3\right)\cdot 17^{4} +O(17^{5})$$ 7*a^4 + 11*a^3 + 3*a^2 + 9*a + 1 + (10*a^4 + 13*a^3 + 3*a^2 + 15*a + 3)*17 + (11*a^4 + 9*a^3 + 12*a^2 + 8*a + 8)*17^2 + (14*a^4 + 12*a^3 + 16*a^2 + 13*a + 2)*17^3 + (15*a^4 + 4*a^3 + 5*a^2 + a + 3)*17^4+O(17^5) $r_{ 9 }$ $=$ $$10 a^{4} + 8 a^{3} + 15 a^{2} + a + \left(11 a^{4} + 14 a^{3} + 4 a^{2} + 10 a + 4\right)\cdot 17 + \left(6 a^{4} + 9 a^{3} + 8 a^{2} + 12 a + 4\right)\cdot 17^{2} + \left(9 a^{4} + 9 a^{3} + 14 a^{2} + 16 a + 15\right)\cdot 17^{3} + \left(2 a^{4} + a^{3} + 15 a^{2} + a + 12\right)\cdot 17^{4} +O(17^{5})$$ 10*a^4 + 8*a^3 + 15*a^2 + a + (11*a^4 + 14*a^3 + 4*a^2 + 10*a + 4)*17 + (6*a^4 + 9*a^3 + 8*a^2 + 12*a + 4)*17^2 + (9*a^4 + 9*a^3 + 14*a^2 + 16*a + 15)*17^3 + (2*a^4 + a^3 + 15*a^2 + a + 12)*17^4+O(17^5) $r_{ 10 }$ $=$ $$10 a^{4} + 14 a^{3} + 6 a^{2} + 16 a + \left(a^{4} + 7 a^{3} + 7 a^{2} + 11 a + 13\right)\cdot 17 + \left(15 a^{4} + 9 a^{3} + 5 a^{2} + 11 a\right)\cdot 17^{2} + \left(10 a^{4} + 5 a^{3} + 8 a^{2} + 12 a + 13\right)\cdot 17^{3} + \left(6 a^{4} + 8 a^{3} + 12 a^{2} + 12\right)\cdot 17^{4} +O(17^{5})$$ 10*a^4 + 14*a^3 + 6*a^2 + 16*a + (a^4 + 7*a^3 + 7*a^2 + 11*a + 13)*17 + (15*a^4 + 9*a^3 + 5*a^2 + 11*a)*17^2 + (10*a^4 + 5*a^3 + 8*a^2 + 12*a + 13)*17^3 + (6*a^4 + 8*a^3 + 12*a^2 + 12)*17^4+O(17^5)

### Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

 Cycle notation $(1,5,2,8,6,3,4,10,7,9)$ $(1,3)(2,10)(4,5)(6,9)(7,8)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character values $c1$ $c2$ $c3$ $c4$ $1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $2$ $(1,3)(2,10)(4,5)(6,9)(7,8)$ $-1$ $-1$ $-1$ $-1$ $1$ $5$ $(1,2,6,4,7)(3,10,9,5,8)$ $\zeta_{5}$ $\zeta_{5}^{2}$ $\zeta_{5}^{3}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,6,7,2,4)(3,9,8,10,5)$ $\zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}$ $\zeta_{5}^{3}$ $1$ $5$ $(1,4,2,7,6)(3,5,10,8,9)$ $\zeta_{5}^{3}$ $\zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{2}$ $1$ $5$ $(1,7,4,6,2)(3,8,5,9,10)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{3}$ $\zeta_{5}^{2}$ $\zeta_{5}$ $1$ $10$ $(1,5,2,8,6,3,4,10,7,9)$ $-\zeta_{5}^{3}$ $-\zeta_{5}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $-\zeta_{5}^{2}$ $1$ $10$ $(1,8,4,9,2,3,7,5,6,10)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $-\zeta_{5}^{3}$ $-\zeta_{5}^{2}$ $-\zeta_{5}$ $1$ $10$ $(1,10,6,5,7,3,2,9,4,8)$ $-\zeta_{5}$ $-\zeta_{5}^{2}$ $-\zeta_{5}^{3}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $1$ $10$ $(1,9,7,10,4,3,6,8,2,5)$ $-\zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ $-\zeta_{5}$ $-\zeta_{5}^{3}$
The blue line marks the conjugacy class containing complex conjugation.