# Properties

 Label 2.10575.10t3.b.b Dimension $2$ Group $D_{10}$ Conductor $10575$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{10}$ Conductor: $$10575$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 47$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 10.2.174158864690625.1 Galois orbit size: $2$ Smallest permutation container: $D_{10}$ Parity: odd Determinant: 1.47.2t1.a.a Projective image: $D_5$ Projective stem field: 5.1.2209.1

## Defining polynomial

 $f(x)$ $=$ $$x^{10} - 4 x^{9} - 2 x^{8} - 10 x^{7} + 207 x^{6} - 172 x^{5} - 481 x^{4} - 1904 x^{3} + 2339 x^{2} - 11530 x - 24155$$  .

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.