# Properties

 Label 1.45.12t1.a.b Dimension $1$ Group $C_{12}$ Conductor $45$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{12}$ Conductor: $$45$$$$\medspace = 3^{2} \cdot 5$$ Artin field: Galois closure of 12.0.84075626953125.1 Galois orbit size: $4$ Smallest permutation container: $C_{12}$ Parity: odd Dirichlet character: $$\chi_{45}(43,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{12} + 3x^{10} - x^{9} + 9x^{8} + 9x^{7} + 28x^{6} + 18x^{5} + 75x^{4} + 26x^{3} + 9x^{2} + 3x + 1$$ x^12 + 3*x^10 - x^9 + 9*x^8 + 9*x^7 + 28*x^6 + 18*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{4} + 7x^{2} + 10x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$a^{3} + 16 a^{2} + 5 + \left(13 a + 10\right)\cdot 17 + \left(9 a^{3} + 12 a^{2} + 9 a + 2\right)\cdot 17^{2} + \left(4 a^{3} + 6 a^{2} + 5 a + 9\right)\cdot 17^{3} + \left(4 a^{2} + 4 a + 8\right)\cdot 17^{4} +O(17^{5})$$ a^3 + 16*a^2 + 5 + (13*a + 10)*17 + (9*a^3 + 12*a^2 + 9*a + 2)*17^2 + (4*a^3 + 6*a^2 + 5*a + 9)*17^3 + (4*a^2 + 4*a + 8)*17^4+O(17^5) $r_{ 2 }$ $=$ $$12 a^{3} + a^{2} + 5 a + 11 + \left(5 a^{3} + 10 a^{2} + 5 a + 4\right)\cdot 17 + \left(3 a^{3} + 12 a^{2} + 7 a + 15\right)\cdot 17^{2} + \left(7 a^{3} + 14 a^{2} + 9 a + 9\right)\cdot 17^{3} + \left(11 a^{3} + 8 a^{2} + 13 a + 9\right)\cdot 17^{4} +O(17^{5})$$ 12*a^3 + a^2 + 5*a + 11 + (5*a^3 + 10*a^2 + 5*a + 4)*17 + (3*a^3 + 12*a^2 + 7*a + 15)*17^2 + (7*a^3 + 14*a^2 + 9*a + 9)*17^3 + (11*a^3 + 8*a^2 + 13*a + 9)*17^4+O(17^5) $r_{ 3 }$ $=$ $$8 a^{3} + 11 a^{2} + 11 a + 10 + \left(14 a^{3} + 6 a^{2} + 11\right)\cdot 17 + \left(13 a^{3} + 6 a^{2} + 5 a + 6\right)\cdot 17^{2} + \left(14 a^{3} + 11 a^{2} + 7 a + 14\right)\cdot 17^{3} + \left(13 a^{3} + 12 a^{2} + a\right)\cdot 17^{4} +O(17^{5})$$ 8*a^3 + 11*a^2 + 11*a + 10 + (14*a^3 + 6*a^2 + 11)*17 + (13*a^3 + 6*a^2 + 5*a + 6)*17^2 + (14*a^3 + 11*a^2 + 7*a + 14)*17^3 + (13*a^3 + 12*a^2 + a)*17^4+O(17^5) $r_{ 4 }$ $=$ $$4 a^{3} + 14 a^{2} + 14 a + 5 + \left(13 a^{3} + 15 a^{2} + 12 a + 13\right)\cdot 17 + \left(11 a^{3} + 16 a^{2} + 11 a + 8\right)\cdot 17^{2} + \left(5 a^{3} + 15 a^{2} + 4 a + 11\right)\cdot 17^{3} + \left(a^{2} + 13 a + 12\right)\cdot 17^{4} +O(17^{5})$$ 4*a^3 + 14*a^2 + 14*a + 5 + (13*a^3 + 15*a^2 + 12*a + 13)*17 + (11*a^3 + 16*a^2 + 11*a + 8)*17^2 + (5*a^3 + 15*a^2 + 4*a + 11)*17^3 + (a^2 + 13*a + 12)*17^4+O(17^5) $r_{ 5 }$ $=$ $$5 a^{3} + 12 a^{2} + 8 + \left(4 a^{2} + 14 a\right)\cdot 17 + \left(2 a^{3} + a^{2} + 14 a + 2\right)\cdot 17^{2} + \left(16 a^{3} + 14 a^{2} + 12 a + 8\right)\cdot 17^{3} + \left(13 a^{3} + 8 a + 2\right)\cdot 17^{4} +O(17^{5})$$ 5*a^3 + 12*a^2 + 8 + (4*a^2 + 14*a)*17 + (2*a^3 + a^2 + 14*a + 2)*17^2 + (16*a^3 + 14*a^2 + 12*a + 8)*17^3 + (13*a^3 + 8*a + 2)*17^4+O(17^5) $r_{ 6 }$ $=$ $$13 a^{3} + 12 a^{2} + 6 a + 13 + \left(8 a^{3} + 14 a^{2} + a + 4\right)\cdot 17 + \left(14 a^{3} + 8 a + 13\right)\cdot 17^{2} + \left(7 a^{3} + 13 a^{2} + 4 a + 13\right)\cdot 17^{3} + \left(13 a^{3} + a^{2} + 15 a + 4\right)\cdot 17^{4} +O(17^{5})$$ 13*a^3 + 12*a^2 + 6*a + 13 + (8*a^3 + 14*a^2 + a + 4)*17 + (14*a^3 + 8*a + 13)*17^2 + (7*a^3 + 13*a^2 + 4*a + 13)*17^3 + (13*a^3 + a^2 + 15*a + 4)*17^4+O(17^5) $r_{ 7 }$ $=$ $$11 a^{3} + 6 a^{2} + 4 + \left(16 a^{3} + 11 a^{2} + 7 a + 6\right)\cdot 17 + \left(5 a^{3} + 3 a^{2} + 9 a + 12\right)\cdot 17^{2} + \left(13 a^{3} + 13 a^{2} + 15 a + 16\right)\cdot 17^{3} + \left(2 a^{3} + 11 a^{2} + 3 a + 5\right)\cdot 17^{4} +O(17^{5})$$ 11*a^3 + 6*a^2 + 4 + (16*a^3 + 11*a^2 + 7*a + 6)*17 + (5*a^3 + 3*a^2 + 9*a + 12)*17^2 + (13*a^3 + 13*a^2 + 15*a + 16)*17^3 + (2*a^3 + 11*a^2 + 3*a + 5)*17^4+O(17^5) $r_{ 8 }$ $=$ $$15 a^{3} + 14 a^{2} + 2 a + 1 + \left(a^{3} + 6 a^{2} + 16 a + 2\right)\cdot 17 + \left(2 a^{3} + 10 a^{2} + 15 a + 13\right)\cdot 17^{2} + \left(8 a^{3} + 7 a^{2} + a + 13\right)\cdot 17^{3} + \left(8 a^{2} + 12 a + 15\right)\cdot 17^{4} +O(17^{5})$$ 15*a^3 + 14*a^2 + 2*a + 1 + (a^3 + 6*a^2 + 16*a + 2)*17 + (2*a^3 + 10*a^2 + 15*a + 13)*17^2 + (8*a^3 + 7*a^2 + a + 13)*17^3 + (8*a^2 + 12*a + 15)*17^4+O(17^5) $r_{ 9 }$ $=$ $$14 a^{3} + 9 a^{2} + 13 a + 14 + \left(9 a^{3} + 5 a^{2} + 6 a + 6\right)\cdot 17 + \left(6 a^{3} + 15 a^{2} + 3 a\right)\cdot 17^{2} + \left(a^{3} + 13 a^{2} + 8 a + 16\right)\cdot 17^{3} + \left(a^{3} + 3 a^{2} + 15 a + 9\right)\cdot 17^{4} +O(17^{5})$$ 14*a^3 + 9*a^2 + 13*a + 14 + (9*a^3 + 5*a^2 + 6*a + 6)*17 + (6*a^3 + 15*a^2 + 3*a)*17^2 + (a^3 + 13*a^2 + 8*a + 16)*17^3 + (a^3 + 3*a^2 + 15*a + 9)*17^4+O(17^5) $r_{ 10 }$ $=$ $$5 a^{3} + 9 a^{2} + 9 a + 2 + \left(6 a^{3} + 11 a^{2} + 3 a + 9\right)\cdot 17 + \left(8 a^{3} + 10 a^{2} + 1\right)\cdot 17^{2} + \left(13 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 17^{3} + \left(2 a^{3} + 2 a^{2} + 2 a + 3\right)\cdot 17^{4} +O(17^{5})$$ 5*a^3 + 9*a^2 + 9*a + 2 + (6*a^3 + 11*a^2 + 3*a + 9)*17 + (8*a^3 + 10*a^2 + 1)*17^2 + (13*a^3 + 6*a^2 + 5*a + 8)*17^3 + (2*a^3 + 2*a^2 + 2*a + 3)*17^4+O(17^5) $r_{ 11 }$ $=$ $$7 a^{3} + 13 a^{2} + 15 a + 7 + \left(15 a^{3} + 13 a^{2} + 8 a + 5\right)\cdot 17 + \left(12 a^{3} + 5 a + 3\right)\cdot 17^{2} + \left(7 a^{3} + 7 a^{2} + 4 a + 4\right)\cdot 17^{3} + \left(2 a^{3} + 11 a^{2} + 3 a + 2\right)\cdot 17^{4} +O(17^{5})$$ 7*a^3 + 13*a^2 + 15*a + 7 + (15*a^3 + 13*a^2 + 8*a + 5)*17 + (12*a^3 + 5*a + 3)*17^2 + (7*a^3 + 7*a^2 + 4*a + 4)*17^3 + (2*a^3 + 11*a^2 + 3*a + 2)*17^4+O(17^5) $r_{ 12 }$ $=$ $$7 a^{3} + 2 a^{2} + 10 a + 5 + \left(9 a^{3} + 12 a + 10\right)\cdot 17 + \left(11 a^{3} + 11 a^{2} + 10 a + 5\right)\cdot 17^{2} + \left(a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 17^{3} + \left(5 a^{3} + 16 a^{2} + 8 a + 8\right)\cdot 17^{4} +O(17^{5})$$ 7*a^3 + 2*a^2 + 10*a + 5 + (9*a^3 + 12*a + 10)*17 + (11*a^3 + 11*a^2 + 10*a + 5)*17^2 + (a^3 + 11*a^2 + 5*a + 10)*17^3 + (5*a^3 + 16*a^2 + 8*a + 8)*17^4+O(17^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

 Cycle notation $(1,10,6,8)(2,7,4,11)(3,9,12,5)$ $(1,11,5,6,7,9)(2,3,8,4,12,10)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,6)(2,4)(3,12)(5,9)(7,11)(8,10)$ $-1$ $1$ $3$ $(1,5,7)(2,8,12)(3,4,10)(6,9,11)$ $-\zeta_{12}^{2}$ $1$ $3$ $(1,7,5)(2,12,8)(3,10,4)(6,11,9)$ $\zeta_{12}^{2} - 1$ $1$ $4$ $(1,10,6,8)(2,7,4,11)(3,9,12,5)$ $-\zeta_{12}^{3}$ $1$ $4$ $(1,8,6,10)(2,11,4,7)(3,5,12,9)$ $\zeta_{12}^{3}$ $1$ $6$ $(1,11,5,6,7,9)(2,3,8,4,12,10)$ $-\zeta_{12}^{2} + 1$ $1$ $6$ $(1,9,7,6,5,11)(2,10,12,4,8,3)$ $\zeta_{12}^{2}$ $1$ $12$ $(1,2,9,10,7,12,6,4,5,8,11,3)$ $-\zeta_{12}$ $1$ $12$ $(1,12,11,10,5,2,6,3,7,8,9,4)$ $-\zeta_{12}^{3} + \zeta_{12}$ $1$ $12$ $(1,4,9,8,7,3,6,2,5,10,11,12)$ $\zeta_{12}$ $1$ $12$ $(1,3,11,8,5,4,6,12,7,10,9,2)$ $\zeta_{12}^{3} - \zeta_{12}$

The blue line marks the conjugacy class containing complex conjugation.