# Properties

 Label 1.183.10t1.a.a Dimension $1$ Group $C_{10}$ Conductor $183$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $$183$$$$\medspace = 3 \cdot 61$$ Artin field: 10.0.46584877058339283.1 Galois orbit size: $4$ Smallest permutation container: $C_{10}$ Parity: odd Dirichlet character: $$\chi_{183}(20,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{10} - x^{9} + 25 x^{8} - 10 x^{7} + 552 x^{6} - 313 x^{5} + 1286 x^{4} + 1321 x^{3} + 1460 x^{2} + 533 x + 169$$  .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{5} + 5 x + 17$$

Roots:
 $r_{ 1 }$ $=$ $$4 a^{4} + 13 a^{3} + 6 a^{2} + 7 + \left(17 a^{4} + a^{3} + 18 a^{2} + 5 a + 16\right)\cdot 19 + \left(14 a^{4} + 11 a^{3} + 14 a^{2} + a + 4\right)\cdot 19^{2} + \left(8 a^{4} + 17 a^{3} + 16 a^{2} + 4 a + 12\right)\cdot 19^{3} + \left(6 a^{4} + 10 a^{3} + 11 a^{2} + a + 6\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 2 }$ $=$ $$5 a^{4} + 4 a^{3} + 10 a^{2} + 10 a + 14 + \left(14 a^{4} + 12 a^{3} + 10 a + 10\right)\cdot 19 + \left(13 a^{4} + 4 a^{3} + 10 a^{2} + 2 a + 18\right)\cdot 19^{2} + \left(18 a^{4} + 9 a^{3} + 6 a^{2} + 18 a + 17\right)\cdot 19^{3} + \left(14 a^{4} + 6 a^{3} + 9 a^{2} + 14 a + 6\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 3 }$ $=$ $$8 a^{4} + 11 a^{3} + 10 a^{2} + 12 a + 4 + \left(13 a^{4} + 3 a^{3} + 11 a^{2} + 14 a + 1\right)\cdot 19 + \left(8 a^{4} + 2 a^{3} + 5 a^{2} + 6 a + 18\right)\cdot 19^{2} + \left(8 a^{4} + 8 a^{3} + 13 a + 10\right)\cdot 19^{3} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 7 a + 13\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 4 }$ $=$ $$8 a^{4} + 15 a^{3} + 14 a^{2} + 16 a + 7 + \left(2 a^{3} + 14 a^{2} + 10 a + 12\right)\cdot 19 + \left(2 a^{4} + 8 a^{3} + 5 a + 9\right)\cdot 19^{2} + \left(12 a^{4} + 6 a^{3} + 10 a^{2} + 13 a + 10\right)\cdot 19^{3} + \left(14 a^{4} + 13 a^{2} + a + 5\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 5 }$ $=$ $$9 a^{4} + 15 a^{3} + 4 a^{2} + 11 + \left(9 a^{4} + 16 a^{3} + 9 a^{2} + 16 a + 10\right)\cdot 19 + \left(10 a^{4} + 3 a^{3} + 3 a^{2} + 7 a + 5\right)\cdot 19^{2} + \left(6 a^{4} + 16 a^{3} + 14 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(12 a^{3} + 8 a^{2} + 2 a + 5\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 6 }$ $=$ $$12 a^{4} + 13 a^{3} + 2 a^{2} + 5 a + 1 + \left(9 a^{4} + 18 a^{3} + 14 a^{2} + 15 a + 5\right)\cdot 19 + \left(16 a^{4} + 15 a^{3} + 16 a^{2} + 3 a + 11\right)\cdot 19^{2} + \left(2 a^{4} + 15 a^{3} + 13 a^{2} + 7\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 6 a^{2} + 17 a + 8\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 7 }$ $=$ $$16 a^{4} + 14 a^{3} + 5 a^{2} + 6 a + 17 + \left(15 a^{4} + 15 a^{2} + 9 a + 10\right)\cdot 19 + \left(14 a^{4} + 10 a^{3} + 6 a^{2} + 8 a + 4\right)\cdot 19^{2} + \left(17 a^{4} + 16 a^{3} + 4 a^{2} + 11 a + 10\right)\cdot 19^{3} + \left(14 a^{4} + 10 a^{3} + 14 a^{2} + 13 a + 2\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 8 }$ $=$ $$17 a^{4} + 3 a^{3} + 16 a^{2} + 4 a + 5 + \left(8 a^{4} + 6 a^{3} + 10 a^{2} + 3 a + 8\right)\cdot 19 + \left(16 a^{4} + 8 a^{3} + 7 a^{2} + 13 a + 10\right)\cdot 19^{2} + \left(a^{4} + 5 a^{3} + 11 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(16 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 11\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 9 }$ $=$ $$17 a^{4} + 6 a^{3} + 15 a^{2} + 15 a + 2 + \left(13 a^{3} + 16 a^{2} + 12 a + 8\right)\cdot 19 + \left(2 a^{4} + 17 a^{3} + 12 a^{2} + 17 a + 10\right)\cdot 19^{2} + \left(17 a^{3} + 2 a^{2} + 8 a + 15\right)\cdot 19^{3} + \left(11 a^{4} + 13 a^{2} + 17 a + 5\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 10 }$ $=$ $$18 a^{4} + a^{3} + 13 a^{2} + 8 a + 9 + \left(4 a^{4} + 2 a^{2} + 16 a + 11\right)\cdot 19 + \left(14 a^{4} + 13 a^{3} + 16 a^{2} + 8 a + 1\right)\cdot 19^{2} + \left(17 a^{4} + 14 a^{2} + 12 a + 14\right)\cdot 19^{3} + \left(10 a^{4} + 9 a^{3} + 16 a^{2} + 9 a + 9\right)\cdot 19^{4} +O(19^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.