# Properties

 Label 16.327...881.36t1252.a Dimension $16$ Group $S_6$ Conductor $3.277\times 10^{34}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $16$ Group: $S_6$ Conductor: $$327\!\cdots\!881$$$$\medspace = 20627^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.20627.1 Galois orbit size: $1$ Smallest permutation container: 36T1252 Parity: even Projective image: $S_6$ Projective field: 6.0.20627.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $$x^{2} + 274 x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$62 a + 226 + \left(43 a + 116\right)\cdot 277 + \left(152 a + 9\right)\cdot 277^{2} + \left(231 a + 84\right)\cdot 277^{3} + \left(175 a + 85\right)\cdot 277^{4} +O(277^{5})$$ $r_{ 2 }$ $=$ $$230 + 135\cdot 277 + 119\cdot 277^{2} + 146\cdot 277^{3} + 55\cdot 277^{4} +O(277^{5})$$ $r_{ 3 }$ $=$ $$118 + 184\cdot 277 + 194\cdot 277^{2} + 65\cdot 277^{3} + 153\cdot 277^{4} +O(277^{5})$$ $r_{ 4 }$ $=$ $$41 + 146\cdot 277 + 30\cdot 277^{2} + 9\cdot 277^{3} + 24\cdot 277^{4} +O(277^{5})$$ $r_{ 5 }$ $=$ $$215 a + 135 + \left(233 a + 184\right)\cdot 277 + \left(124 a + 145\right)\cdot 277^{2} + \left(45 a + 72\right)\cdot 277^{3} + \left(101 a + 104\right)\cdot 277^{4} +O(277^{5})$$ $r_{ 6 }$ $=$ $$82 + 63\cdot 277 + 54\cdot 277^{2} + 176\cdot 277^{3} + 131\cdot 277^{4} +O(277^{5})$$

### 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.