# Properties

 Label 3.2e2_5_443.6t11.2c1 Dimension 3 Group $S_4\times C_2$ Conductor $2^{2} \cdot 5 \cdot 443$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $8860= 2^{2} \cdot 5 \cdot 443$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - 5 x^{4} + 6 x^{3} + 22 x^{2} - 8 x - 32$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4\times C_2$ Parity: Even Determinant: 1.2e2_5_443.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$
Roots:
 $r_{ 1 }$ $=$ $21 + 4\cdot 41^{2} + 40\cdot 41^{3} + 25\cdot 41^{4} + 23\cdot 41^{5} + 31\cdot 41^{6} + 41^{7} + 4\cdot 41^{8} +O\left(41^{ 9 }\right)$ $r_{ 2 }$ $=$ $17 a + 1 + \left(18 a + 18\right)\cdot 41 + \left(31 a + 6\right)\cdot 41^{2} + \left(33 a + 14\right)\cdot 41^{3} + \left(22 a + 38\right)\cdot 41^{4} + \left(8 a + 17\right)\cdot 41^{5} + \left(25 a + 30\right)\cdot 41^{6} + \left(35 a + 36\right)\cdot 41^{7} + \left(38 a + 28\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ $r_{ 3 }$ $=$ $26 + 22\cdot 41 + 39\cdot 41^{2} + 31\cdot 41^{3} + 38\cdot 41^{4} + 20\cdot 41^{5} + 37\cdot 41^{6} + 23\cdot 41^{7} + 28\cdot 41^{8} +O\left(41^{ 9 }\right)$ $r_{ 4 }$ $=$ $28 a + 32 + \left(10 a + 10\right)\cdot 41 + \left(10 a + 26\right)\cdot 41^{2} + \left(27 a + 22\right)\cdot 41^{3} + \left(38 a + 31\right)\cdot 41^{4} + \left(10 a + 22\right)\cdot 41^{5} + \left(22 a + 37\right)\cdot 41^{6} + \left(38 a + 26\right)\cdot 41^{7} + \left(22 a + 1\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ $r_{ 5 }$ $=$ $24 a + 11 + \left(22 a + 15\right)\cdot 41 + 9 a\cdot 41^{2} + \left(7 a + 2\right)\cdot 41^{3} + \left(18 a + 32\right)\cdot 41^{4} + \left(32 a + 20\right)\cdot 41^{5} + \left(15 a + 15\right)\cdot 41^{6} + \left(5 a + 36\right)\cdot 41^{7} + \left(2 a + 27\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$ $r_{ 6 }$ $=$ $13 a + 34 + \left(30 a + 14\right)\cdot 41 + \left(30 a + 5\right)\cdot 41^{2} + \left(13 a + 12\right)\cdot 41^{3} + \left(2 a + 38\right)\cdot 41^{4} + \left(30 a + 16\right)\cdot 41^{5} + \left(18 a + 11\right)\cdot 41^{6} + \left(2 a + 38\right)\cdot 41^{7} + \left(18 a + 31\right)\cdot 41^{8} +O\left(41^{ 9 }\right)$

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

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

### Character values on conjugacy classes

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