# Properties

 Label 1.180.12t1.b.b Dimension $1$ Group $C_{12}$ Conductor $180$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{12}$ Conductor: $$180$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 5$$ Artin field: 12.12.344373768000000000.1 Galois orbit size: $4$ Smallest permutation container: $C_{12}$ Parity: even Dirichlet character: $$\chi_{180}(43,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - 27 x^{10} - 4 x^{9} + 234 x^{8} + 36 x^{7} - 737 x^{6} + 72 x^{5} + 795 x^{4} - 336 x^{3} - 96 x^{2} + 42 x + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.