# Properties

 Label 1.455.12t1.a.b Dimension $1$ Group $C_{12}$ Conductor $455$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{12}$ Conductor: $$455$$$$\medspace = 5 \cdot 7 \cdot 13$$ Artin field: 12.12.187441217958845703125.1 Galois orbit size: $4$ Smallest permutation container: $C_{12}$ Parity: even Dirichlet character: $$\chi_{455}(412,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - x^{11} - 45 x^{10} + 15 x^{9} + 661 x^{8} - 70 x^{7} - 3836 x^{6} + 660 x^{5} + 7876 x^{4} - 3890 x^{3} - 1930 x^{2} + 1489 x - 229$$  .

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $$x^{4} + 8 x^{2} + 40 x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$25 a^{3} + 34 a^{2} + 46 a + 37 + \left(19 a^{3} + 39 a^{2} + 43 a + 15\right)\cdot 47 + \left(8 a^{3} + 46 a^{2} + 6 a + 32\right)\cdot 47^{2} + \left(29 a^{3} + 2 a^{2} + 14 a + 23\right)\cdot 47^{3} + \left(4 a^{3} + 7 a^{2} + 36 a + 21\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 2 }$ $=$ $$25 a^{3} + 34 a^{2} + 46 a + 42 + \left(19 a^{3} + 39 a^{2} + 43 a + 40\right)\cdot 47 + \left(8 a^{3} + 46 a^{2} + 6 a + 44\right)\cdot 47^{2} + \left(29 a^{3} + 2 a^{2} + 14 a + 37\right)\cdot 47^{3} + \left(4 a^{3} + 7 a^{2} + 36 a + 25\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 3 }$ $=$ $$25 a^{3} + 34 a^{2} + 46 a + 6 + \left(19 a^{3} + 39 a^{2} + 43 a + 4\right)\cdot 47 + \left(8 a^{3} + 46 a^{2} + 6 a + 32\right)\cdot 47^{2} + \left(29 a^{3} + 2 a^{2} + 14 a + 3\right)\cdot 47^{3} + \left(4 a^{3} + 7 a^{2} + 36 a + 42\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 4 }$ $=$ $$32 a^{3} + 15 a^{2} + 15 a + 46 + \left(15 a^{3} + 22 a^{2} + 3 a + 6\right)\cdot 47 + \left(23 a^{3} + 19 a^{2} + 26 a + 41\right)\cdot 47^{2} + \left(17 a^{3} + 40 a^{2} + 40 a + 37\right)\cdot 47^{3} + \left(6 a^{3} + 8 a^{2} + 42 a + 7\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 5 }$ $=$ $$32 a^{3} + 15 a^{2} + 15 a + 30 + \left(15 a^{3} + 22 a^{2} + 3 a + 18\right)\cdot 47 + \left(23 a^{3} + 19 a^{2} + 26 a + 41\right)\cdot 47^{2} + \left(17 a^{3} + 40 a^{2} + 40 a + 10\right)\cdot 47^{3} + \left(6 a^{3} + 8 a^{2} + 42 a + 34\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 6 }$ $=$ $$32 a^{3} + 15 a^{2} + 15 a + 35 + \left(15 a^{3} + 22 a^{2} + 3 a + 43\right)\cdot 47 + \left(23 a^{3} + 19 a^{2} + 26 a + 6\right)\cdot 47^{2} + \left(17 a^{3} + 40 a^{2} + 40 a + 25\right)\cdot 47^{3} + \left(6 a^{3} + 8 a^{2} + 42 a + 38\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 7 }$ $=$ $$31 a^{3} + 30 a^{2} + 17 a + 13 + \left(35 a^{3} + 46 a^{2} + 26 a + 10\right)\cdot 47 + \left(36 a^{3} + 34 a^{2} + 23 a + 36\right)\cdot 47^{2} + \left(19 a^{3} + 25 a^{2} + 44 a + 20\right)\cdot 47^{3} + \left(39 a^{3} + 2 a^{2} + 13\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 8 }$ $=$ $$31 a^{3} + 30 a^{2} + 17 a + 18 + \left(35 a^{3} + 46 a^{2} + 26 a + 35\right)\cdot 47 + \left(36 a^{3} + 34 a^{2} + 23 a + 1\right)\cdot 47^{2} + \left(19 a^{3} + 25 a^{2} + 44 a + 35\right)\cdot 47^{3} + \left(39 a^{3} + 2 a^{2} + 17\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 9 }$ $=$ $$31 a^{3} + 30 a^{2} + 17 a + 29 + \left(35 a^{3} + 46 a^{2} + 26 a + 45\right)\cdot 47 + \left(36 a^{3} + 34 a^{2} + 23 a + 35\right)\cdot 47^{2} + \left(19 a^{3} + 25 a^{2} + 44 a\right)\cdot 47^{3} + \left(39 a^{3} + 2 a^{2} + 34\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 10 }$ $=$ $$6 a^{3} + 15 a^{2} + 16 a + 2 + \left(23 a^{3} + 32 a^{2} + 20 a\right)\cdot 47 + \left(25 a^{3} + 39 a^{2} + 37 a + 46\right)\cdot 47^{2} + \left(27 a^{3} + 24 a^{2} + 41 a + 14\right)\cdot 47^{3} + \left(43 a^{3} + 28 a^{2} + 13 a + 7\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 11 }$ $=$ $$6 a^{3} + 15 a^{2} + 16 a + 7 + \left(23 a^{3} + 32 a^{2} + 20 a + 25\right)\cdot 47 + \left(25 a^{3} + 39 a^{2} + 37 a + 11\right)\cdot 47^{2} + \left(27 a^{3} + 24 a^{2} + 41 a + 29\right)\cdot 47^{3} + \left(43 a^{3} + 28 a^{2} + 13 a + 11\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 12 }$ $=$ $$6 a^{3} + 15 a^{2} + 16 a + 18 + \left(23 a^{3} + 32 a^{2} + 20 a + 35\right)\cdot 47 + \left(25 a^{3} + 39 a^{2} + 37 a + 45\right)\cdot 47^{2} + \left(27 a^{3} + 24 a^{2} + 41 a + 41\right)\cdot 47^{3} + \left(43 a^{3} + 28 a^{2} + 13 a + 27\right)\cdot 47^{4} +O(47^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.