# Properties

 Label 9.321...088.10t32.a Dimension $9$ Group $S_6$ Conductor $3.213\times 10^{13}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $9$ Group: $S_6$ Conductor: $$32133168345088$$$$\medspace = 2^{12} \cdot 1987^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.127168.1 Galois orbit size: $1$ Smallest permutation container: $S_{6}$ Parity: even Projective image: $S_6$ Projective field: 6.2.127168.1

## 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 $\mathbb{Q}_{ 29 }$: $$x^{2} + 24 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$13 a + 28 + \left(25 a + 4\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(23 a + 19\right)\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$16 a + 6 + \left(3 a + 3\right)\cdot 29 + \left(10 a + 2\right)\cdot 29^{2} + \left(5 a + 3\right)\cdot 29^{3} + \left(28 a + 5\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$10 a + 26 + \left(4 a + 9\right)\cdot 29 + \left(8 a + 19\right)\cdot 29^{2} + \left(12 a + 19\right)\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$19 a + 18 + \left(24 a + 21\right)\cdot 29 + \left(20 a + 26\right)\cdot 29^{2} + \left(16 a + 14\right)\cdot 29^{3} + \left(28 a + 23\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 5 }$ $=$ $$18 a + 4 + \left(16 a + 20\right)\cdot 29 + \left(21 a + 21\right)\cdot 29^{2} + \left(18 a + 7\right)\cdot 29^{3} + \left(8 a + 2\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 6 }$ $=$ $$11 a + 7 + \left(12 a + 27\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(10 a + 21\right)\cdot 29^{3} + \left(20 a + 26\right)\cdot 29^{4} +O(29^{5})$$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $9$ $15$ $2$ $(1,2)(3,4)(5,6)$ $3$ $15$ $2$ $(1,2)$ $3$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $0$ $40$ $3$ $(1,2,3)$ $0$ $90$ $4$ $(1,2,3,4)(5,6)$ $1$ $90$ $4$ $(1,2,3,4)$ $-1$ $144$ $5$ $(1,2,3,4,5)$ $-1$ $120$ $6$ $(1,2,3,4,5,6)$ $0$ $120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.