# Properties

 Label 16.624...096.36t1252.b Dimension $16$ Group $S_6$ Conductor $6.241\times 10^{24}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $16$ Group: $S_6$ Conductor: $$624\!\cdots\!096$$$$\medspace = 2^{36} \cdot 3^{8} \cdot 7^{12}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.3687936.1 Galois orbit size: $1$ Smallest permutation container: 36T1252 Parity: even Projective image: $S_6$ Projective field: Galois closure of 6.2.3687936.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $$x^{2} + 192x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$167 a + 52 + \left(186 a + 78\right)\cdot 193 + \left(190 a + 25\right)\cdot 193^{2} + \left(157 a + 90\right)\cdot 193^{3} + \left(112 a + 99\right)\cdot 193^{4} +O(193^{5})$$ 167*a + 52 + (186*a + 78)*193 + (190*a + 25)*193^2 + (157*a + 90)*193^3 + (112*a + 99)*193^4+O(193^5) $r_{ 2 }$ $=$ $$49 + 23\cdot 193 + 35\cdot 193^{2} + 151\cdot 193^{3} + 93\cdot 193^{4} +O(193^{5})$$ 49 + 23*193 + 35*193^2 + 151*193^3 + 93*193^4+O(193^5) $r_{ 3 }$ $=$ $$26 a + 26 + \left(6 a + 98\right)\cdot 193 + \left(2 a + 29\right)\cdot 193^{2} + \left(35 a + 57\right)\cdot 193^{3} + \left(80 a + 54\right)\cdot 193^{4} +O(193^{5})$$ 26*a + 26 + (6*a + 98)*193 + (2*a + 29)*193^2 + (35*a + 57)*193^3 + (80*a + 54)*193^4+O(193^5) $r_{ 4 }$ $=$ $$62 + 104\cdot 193 + 147\cdot 193^{2} + 13\cdot 193^{3} + 32\cdot 193^{4} +O(193^{5})$$ 62 + 104*193 + 147*193^2 + 13*193^3 + 32*193^4+O(193^5) $r_{ 5 }$ $=$ $$134 a + 128 + \left(57 a + 175\right)\cdot 193 + \left(141 a + 128\right)\cdot 193^{2} + \left(69 a + 72\right)\cdot 193^{3} + \left(149 a + 13\right)\cdot 193^{4} +O(193^{5})$$ 134*a + 128 + (57*a + 175)*193 + (141*a + 128)*193^2 + (69*a + 72)*193^3 + (149*a + 13)*193^4+O(193^5) $r_{ 6 }$ $=$ $$59 a + 69 + \left(135 a + 99\right)\cdot 193 + \left(51 a + 19\right)\cdot 193^{2} + \left(123 a + 1\right)\cdot 193^{3} + \left(43 a + 93\right)\cdot 193^{4} +O(193^{5})$$ 59*a + 69 + (135*a + 99)*193 + (51*a + 19)*193^2 + (123*a + 1)*193^3 + (43*a + 93)*193^4+O(193^5)

### 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 $c1$ $1$ $1$ $()$ $16$ $15$ $2$ $(1,2)(3,4)(5,6)$ $0$ $15$ $2$ $(1,2)$ $0$ $45$ $2$ $(1,2)(3,4)$ $0$ $40$ $3$ $(1,2,3)(4,5,6)$ $-2$ $40$ $3$ $(1,2,3)$ $-2$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $90$ $4$ $(1,2,3,4)$ $0$ $144$ $5$ $(1,2,3,4,5)$ $1$ $120$ $6$ $(1,2,3,4,5,6)$ $0$ $120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.