# Properties

 Label 3.2083.12t76.a.a Dimension $3$ Group $A_5\times C_2$ Conductor $2083$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5\times C_2$ Conductor: $$2083$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 10.0.39214470002250643.1 Galois orbit size: $2$ Smallest permutation container: 12T76 Parity: odd Determinant: 1.2083.2t1.a.a Projective image: $A_5$ Projective stem field: 5.1.4338889.1

## Defining polynomial

 $f(x)$ $=$ $$x^{10} - 20 x^{7} - 9 x^{6} + 26 x^{5} + 100 x^{4} + 90 x^{3} + 281 x^{2} - 117 x + 169$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{5} + x + 14$$

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

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character value $1$ $1$ $()$ $3$ $1$ $2$ $(1,5)(2,10)(3,7)(4,6)(8,9)$ $-3$ $15$ $2$ $(1,9)(2,4)(3,7)(5,8)(6,10)$ $1$ $15$ $2$ $(1,8)(2,6)(4,10)(5,9)$ $-1$ $20$ $3$ $(1,6,8)(4,9,5)$ $0$ $12$ $5$ $(1,3,8,2,6)(4,5,7,9,10)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $12$ $5$ $(1,2,3,6,8)(4,9,5,10,7)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $20$ $6$ $(1,9,6,5,8,4)(2,10)(3,7)$ $0$ $12$ $10$ $(1,10,3,4,8,5,2,7,6,9)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $12$ $10$ $(1,4,2,9,3,5,6,10,8,7)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.