# Properties

 Label 2.88.10t6.b.d Dimension 2 Group $D_5\times C_5$ Conductor $2^{3} \cdot 11$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_5\times C_5$ Conductor: $88= 2^{3} \cdot 11$ Artin number field: Splitting field of 10.0.479756288.1 defined by $f= x^{10} - 2 x^{9} + x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{4} + 2 x^{3} - x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $D_5\times C_5$ Parity: Odd Determinant: 1.88.10t1.a.d Projective image: $D_5$ Projective field: Galois closure of 5.1.937024.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{5} + x + 14$
Roots:
 $r_{ 1 }$ $=$ $a^{4} + 14 a^{3} + 9 a^{2} + 10 a + 16 + \left(4 a^{3} + 14 a^{2} + 2 a + 13\right)\cdot 17 + \left(10 a^{4} + 7 a^{3} + 10 a^{2} + 12 a + 13\right)\cdot 17^{2} + \left(6 a^{3} + 14 a + 10\right)\cdot 17^{3} + \left(8 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 11\right)\cdot 17^{4} + \left(16 a^{4} + 16 a^{3} + 15 a^{2} + 16 a + 16\right)\cdot 17^{5} + \left(14 a^{4} + 6 a^{2} + 4\right)\cdot 17^{6} + \left(16 a^{4} + 12 a^{3} + 5 a^{2} + 7 a + 15\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 2 }$ $=$ $5 a^{4} + 7 a^{3} + 8 a^{2} + 10 a + 13 + \left(2 a^{3} + 14 a^{2} + 13 a + 6\right)\cdot 17 + \left(12 a^{3} + 11 a^{2} + 13 a + 4\right)\cdot 17^{2} + \left(9 a^{4} + 12 a^{3} + 2 a\right)\cdot 17^{3} + \left(11 a^{4} + a^{3} + 4 a^{2} + 4\right)\cdot 17^{4} + \left(13 a^{3} + 4 a + 7\right)\cdot 17^{5} + \left(14 a^{4} + 4 a^{3} + 2 a^{2} + 14 a + 11\right)\cdot 17^{6} + \left(9 a^{4} + 11 a^{3} + 14 a + 2\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 3 }$ $=$ $6 a^{4} + 9 a^{3} + 3 a^{2} + 4 a + 7 + \left(6 a^{4} + 6 a^{3} + 14 a^{2} + 5 a + 1\right)\cdot 17 + \left(16 a^{4} + 16 a^{2} + 16 a + 14\right)\cdot 17^{2} + \left(9 a^{4} + 14 a^{3} + 13 a^{2} + 15 a + 7\right)\cdot 17^{3} + \left(8 a^{4} + 16 a^{3} + 3 a^{2} + 2 a + 8\right)\cdot 17^{4} + \left(2 a^{4} + a^{3} + 6 a^{2} + 5 a + 15\right)\cdot 17^{5} + \left(2 a^{4} + 2 a^{3} + 2 a^{2} + 16 a + 1\right)\cdot 17^{6} + \left(12 a^{4} + 4 a^{3} + 3 a^{2} + 4 a + 1\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 4 }$ $=$ $8 a^{4} + 11 a^{3} + 6 a^{2} + 12 a + 8 + \left(12 a^{4} + 12 a^{3} + 4 a^{2} + 3 a + 3\right)\cdot 17 + \left(15 a^{4} + 11 a^{3} + 4 a^{2} + 3 a + 8\right)\cdot 17^{2} + \left(2 a^{4} + 16 a^{3} + 4 a^{2} + a + 2\right)\cdot 17^{3} + \left(7 a^{4} + 3 a^{3} + a^{2} + 16 a + 4\right)\cdot 17^{4} + \left(6 a^{4} + 10 a^{3} + 12 a^{2} + 8 a + 5\right)\cdot 17^{5} + \left(9 a^{4} + 16 a^{3} + 6 a^{2} + 6 a + 7\right)\cdot 17^{6} + \left(11 a^{4} + 10 a^{3} + 13 a^{2} + 12 a + 14\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 5 }$ $=$ $9 a^{4} + 16 a^{3} + a^{2} + 8 a + 6 + \left(6 a^{4} + 6 a^{3} + 10 a^{2} + 5 a + 8\right)\cdot 17 + \left(a^{4} + 8 a^{3} + 8 a^{2} + 8 a + 5\right)\cdot 17^{2} + \left(13 a^{4} + 6 a^{3} + 5 a^{2} + 6 a + 10\right)\cdot 17^{3} + \left(11 a^{4} + 9 a^{3} + 16 a^{2} + 8 a + 7\right)\cdot 17^{4} + \left(7 a^{4} + 9 a^{3} + 3 a^{2} + 11 a + 9\right)\cdot 17^{5} + \left(10 a^{4} + 5 a^{3} + 2 a^{2} + 7 a + 8\right)\cdot 17^{6} + \left(4 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 15\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 6 }$ $=$ $12 a^{4} + 7 a^{3} + 16 a^{2} + 16 a + 1 + \left(2 a^{4} + 9 a^{3} + 12 a^{2} + 16 a + 16\right)\cdot 17 + \left(11 a^{4} + 13 a^{3} + 11 a^{2} + 4 a + 7\right)\cdot 17^{2} + \left(10 a^{4} + 14 a^{3} + 10 a^{2} + 14 a + 15\right)\cdot 17^{3} + \left(a^{4} + 11 a^{3} + 6 a^{2} + 3 a + 9\right)\cdot 17^{4} + \left(2 a^{4} + 11 a^{3} + 2 a^{2} + 15 a + 8\right)\cdot 17^{5} + \left(15 a^{4} + 13 a^{3} + 8 a^{2} + 2 a + 8\right)\cdot 17^{6} + \left(4 a^{3} + 7 a^{2} + 16 a + 2\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 7 }$ $=$ $14 a^{4} + 13 a^{3} + 7 a^{2} + 15 a + 6 + \left(6 a^{3} + 5 a^{2} + 7 a + 4\right)\cdot 17 + \left(5 a^{4} + 11 a^{3} + 11 a^{2} + 7 a + 13\right)\cdot 17^{2} + \left(5 a^{3} + 11 a + 3\right)\cdot 17^{3} + \left(8 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 8\right)\cdot 17^{4} + \left(11 a^{4} + 4 a^{3} + 7 a^{2} + 14 a + 2\right)\cdot 17^{5} + \left(3 a^{4} + 8 a^{3} + 15 a + 6\right)\cdot 17^{6} + \left(16 a^{4} + 12 a^{3} + 4 a^{2} + 13 a + 11\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 8 }$ $=$ $15 a^{4} + 14 a^{3} + 5 a^{2} + 4 + \left(16 a^{4} + 4 a^{3} + a^{2} + 15 a + 13\right)\cdot 17 + \left(16 a^{4} + 4 a^{2} + 12 a + 7\right)\cdot 17^{2} + \left(2 a^{4} + 16 a^{3} + 3 a^{2} + a + 12\right)\cdot 17^{3} + \left(8 a^{4} + 12 a^{3} + 16 a^{2} + 12 a + 4\right)\cdot 17^{4} + \left(a^{4} + 11 a^{3} + 12 a^{2} + 11\right)\cdot 17^{5} + \left(2 a^{4} + 12 a^{3} + 14 a^{2} + 13 a + 8\right)\cdot 17^{6} + \left(5 a^{4} + 2 a^{3} + 2 a + 12\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 9 }$ $=$ $16 a^{4} + 5 a^{3} + 12 a + 15 + \left(3 a^{4} + 13 a^{3} + 11 a^{2} + 11 a + 2\right)\cdot 17 + \left(16 a^{4} + 12 a^{3} + 9 a^{2} + 16 a + 7\right)\cdot 17^{2} + \left(15 a^{4} + a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 17^{3} + \left(10 a^{4} + 10 a^{3} + 10 a^{2} + 10 a\right)\cdot 17^{4} + \left(4 a^{4} + 14 a^{3} + 10 a^{2} + 12 a + 7\right)\cdot 17^{5} + \left(5 a^{4} + 8 a^{3} + 12 a^{2} + 16 a + 4\right)\cdot 17^{6} + \left(2 a^{4} + 4 a^{3} + 3 a^{2} + 8 a + 10\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$ $r_{ 10 }$ $=$ $16 a^{4} + 6 a^{3} + 13 a^{2} + 15 a + 11 + \left(13 a^{2} + 2 a + 14\right)\cdot 17 + \left(9 a^{4} + 7 a^{3} + 12 a^{2} + 6 a + 2\right)\cdot 17^{2} + \left(2 a^{4} + 7 a^{3} + 9 a + 2\right)\cdot 17^{3} + \left(9 a^{4} + 15 a^{3} + 8 a^{2} + 16 a + 9\right)\cdot 17^{4} + \left(14 a^{4} + 7 a^{3} + 13 a^{2} + 12 a + 1\right)\cdot 17^{5} + \left(7 a^{4} + 11 a^{3} + 11 a^{2} + 7 a + 6\right)\cdot 17^{6} + \left(5 a^{4} + 10 a^{3} + 3 a^{2} + a + 16\right)\cdot 17^{7} +O\left(17^{ 8 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character value $1$ $1$ $()$ $2$ $5$ $2$ $(1,3)(2,10)(4,9)(5,6)(7,8)$ $0$ $1$ $5$ $(1,6,7,4,10)(2,3,5,8,9)$ $2 \zeta_{5}$ $1$ $5$ $(1,7,10,6,4)(2,5,9,3,8)$ $2 \zeta_{5}^{2}$ $1$ $5$ $(1,4,6,10,7)(2,8,3,9,5)$ $2 \zeta_{5}^{3}$ $1$ $5$ $(1,10,4,7,6)(2,9,8,5,3)$ $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ $2$ $5$ $(2,9,8,5,3)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ $2$ $5$ $(2,8,3,9,5)$ $\zeta_{5}^{3} + 1$ $2$ $5$ $(2,5,9,3,8)$ $\zeta_{5}^{2} + 1$ $2$ $5$ $(2,3,5,8,9)$ $\zeta_{5} + 1$ $2$ $5$ $(1,10,4,7,6)(2,8,3,9,5)$ $-\zeta_{5}^{2} - \zeta_{5} - 1$ $2$ $5$ $(1,4,6,10,7)(2,3,5,8,9)$ $\zeta_{5}^{3} + \zeta_{5}$ $2$ $5$ $(1,7,10,6,4)(2,9,8,5,3)$ $-\zeta_{5}^{3} - \zeta_{5} - 1$ $2$ $5$ $(1,6,7,4,10)(2,5,9,3,8)$ $\zeta_{5}^{2} + \zeta_{5}$ $2$ $5$ $(1,4,6,10,7)(2,5,9,3,8)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $2$ $5$ $(1,6,7,4,10)(2,9,8,5,3)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $5$ $10$ $(1,9,6,2,7,3,4,5,10,8)$ $0$ $5$ $10$ $(1,2,4,8,6,3,10,9,7,5)$ $0$ $5$ $10$ $(1,5,7,9,10,3,6,8,4,2)$ $0$ $5$ $10$ $(1,8,10,5,4,3,7,2,6,9)$ $0$
The blue line marks the conjugacy class containing complex conjugation.