# Properties

 Label 5.37_857.6t16.1c1 Dimension 5 Group $S_6$ Conductor $37 \cdot 857$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $31709= 37 \cdot 857$ Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{3} - 2 x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_6$ Parity: Even Determinant: 1.37_857.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $x^{2} + 70 x + 5$
Roots:
 $r_{ 1 }$ $=$ $39 a + 28 + \left(49 a + 25\right)\cdot 73 + \left(2 a + 71\right)\cdot 73^{2} + \left(48 a + 20\right)\cdot 73^{3} + \left(55 a + 50\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $40 a + 28 + \left(67 a + 43\right)\cdot 73 + 37\cdot 73^{2} + \left(38 a + 30\right)\cdot 73^{3} + \left(49 a + 25\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $34 a + 72 + \left(23 a + 61\right)\cdot 73 + \left(70 a + 29\right)\cdot 73^{2} + \left(24 a + 16\right)\cdot 73^{3} + \left(17 a + 23\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $71 + 6\cdot 73 + 42\cdot 73^{2} + 36\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $33 a + 2 + \left(5 a + 60\right)\cdot 73 + \left(72 a + 45\right)\cdot 73^{2} + \left(34 a + 70\right)\cdot 73^{3} + \left(23 a + 62\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 6 }$ $=$ $19 + 21\cdot 73 + 65\cdot 73^{2} + 43\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$

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