# Properties

 Label 1.77.10t1.a.d Dimension 1 Group $C_{10}$ Conductor $7 \cdot 11$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $77= 7 \cdot 11$ Artin number field: Splitting field of 10.0.3602729712967.1 defined by $f= x^{10} - x^{9} + 14 x^{8} - 7 x^{7} + 85 x^{6} - 29 x^{5} + 218 x^{4} - 8 x^{3} + 216 x^{2} - 48 x + 32$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_{10}$ Parity: Odd Corresponding Dirichlet character: $$\chi_{77}(48,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{5} + 3 x + 27$
Roots:
 $r_{ 1 }$ $=$ $2 a^{4} + 20 a^{3} + 12 a^{2} + 18 a + 18 + \left(15 a^{4} + 26 a^{3} + 12 a^{2} + 16 a\right)\cdot 29 + \left(9 a^{4} + 28 a^{3} + 13 a^{2} + 8 a + 23\right)\cdot 29^{2} + \left(19 a^{4} + a^{3} + 23 a^{2} + 24 a + 2\right)\cdot 29^{3} + \left(a^{4} + 20 a^{3} + 11 a^{2} + 24 a + 5\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 a^{4} + 6 a^{3} + 14 a^{2} + 19 a + 14 + \left(9 a^{4} + 15 a^{3} + 26 a^{2} + 17 a + 28\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 11 a^{2} + 6 a + 15\right)\cdot 29^{2} + \left(26 a^{4} + 5 a^{3} + 17 a^{2} + 8 a + 13\right)\cdot 29^{3} + \left(17 a^{4} + 24 a^{3} + 7 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $5 a^{4} + 10 a^{3} + 13 a^{2} + 24 a + 28 + \left(18 a^{4} + 25 a^{3} + 12 a^{2} + 23 a + 26\right)\cdot 29 + \left(22 a^{4} + 12 a^{3} + 11 a^{2} + 23 a + 1\right)\cdot 29^{2} + \left(14 a^{4} + 7 a^{3} + 15 a^{2} + 6 a + 21\right)\cdot 29^{3} + \left(5 a^{3} + 25 a^{2} + 7 a + 11\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $6 a^{4} + 5 a^{3} + 21 a^{2} + 2 a + 13 + \left(12 a^{4} + 12 a^{3} + 22 a^{2} + 8 a + 18\right)\cdot 29 + \left(11 a^{4} + 9 a^{3} + 27 a^{2} + 3 a + 15\right)\cdot 29^{2} + \left(23 a^{4} + 6 a^{3} + 3 a^{2} + 2 a + 18\right)\cdot 29^{3} + \left(22 a^{4} + 17 a^{3} + 7 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a^{4} + 4 a^{3} + 18 a^{2} + 21 a + 28 + \left(12 a^{4} + 2 a^{3} + 14 a^{2} + 11 a + 28\right)\cdot 29 + \left(19 a^{4} + 18 a^{3} + 19 a^{2} + 10 a + 11\right)\cdot 29^{2} + \left(24 a^{3} + 7 a^{2} + 14 a + 4\right)\cdot 29^{3} + \left(18 a^{4} + 12 a^{3} + 19 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 6 }$ $=$ $12 a^{4} + 18 a^{3} + 13 a^{2} + 28 a + 13 + \left(19 a^{4} + 4 a^{3} + 22 a^{2} + 14 a + 28\right)\cdot 29 + \left(25 a^{4} + 26 a^{3} + 18 a^{2} + 8 a + 26\right)\cdot 29^{2} + \left(26 a^{3} + 23 a^{2} + 8 a + 4\right)\cdot 29^{3} + \left(5 a^{4} + 21 a^{3} + 4 a^{2} + 20 a + 13\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 7 }$ $=$ $15 a^{4} + a^{3} + 10 a^{2} + 14 a + 26 + \left(15 a^{4} + 27 a^{3} + 11 a^{2} + 23 a + 24\right)\cdot 29 + \left(19 a^{4} + 21 a^{3} + a^{2} + 6 a + 23\right)\cdot 29^{2} + \left(28 a^{4} + 6 a^{3} + 19 a^{2} + 22 a + 7\right)\cdot 29^{3} + \left(6 a^{4} + 21 a^{3} + 11 a^{2} + 14 a + 6\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 8 }$ $=$ $18 a^{4} + 15 a^{3} + 5 a^{2} + 6 a + 10 + \left(24 a^{4} + 26 a^{3} + 26 a^{2} + 20 a\right)\cdot 29 + \left(12 a^{4} + 20 a^{3} + 4 a^{2} + 23 a + 2\right)\cdot 29^{2} + \left(8 a^{4} + 26 a^{3} + 13 a^{2} + 17 a + 23\right)\cdot 29^{3} + \left(26 a^{4} + 10 a^{3} + 7 a + 17\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 9 }$ $=$ $20 a^{4} + 26 a^{3} + 4 a^{2} + 6 a + 6 + \left(8 a^{4} + 6 a^{3} + 26 a^{2} + 19 a + 4\right)\cdot 29 + \left(15 a^{4} + a^{2} + 14 a + 19\right)\cdot 29^{2} + \left(26 a^{4} + 26 a^{3} + 5 a^{2} + 5 a + 8\right)\cdot 29^{3} + \left(2 a^{4} + 24 a^{3} + 15 a^{2} + 3 a + 23\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 10 }$ $=$ $23 a^{4} + 11 a^{3} + 6 a^{2} + 7 a + 19 + \left(9 a^{4} + 27 a^{3} + 28 a^{2} + 18 a + 12\right)\cdot 29 + \left(4 a^{4} + 7 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 29^{2} + \left(25 a^{4} + 12 a^{3} + 16 a^{2} + 6 a + 11\right)\cdot 29^{3} + \left(13 a^{4} + 15 a^{3} + 16 a^{2} + 4 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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