# Properties

 Label 5.2e2_71_149.6t16.1 Dimension 5 Group $S_6$ Conductor $2^{2} \cdot 71 \cdot 149$ Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $42316= 2^{2} \cdot 71 \cdot 149$ Artin number field: Splitting field of $f= x^{6} - x^{3} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_6$ Parity: Odd

## 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:
 $r_{ 1 }$ $=$ $12 a + 66 + \left(16 a + 31\right)\cdot 73 + \left(44 a + 20\right)\cdot 73^{2} + \left(34 a + 60\right)\cdot 73^{3} + \left(72 a + 71\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $70 a + 66 + \left(6 a + 52\right)\cdot 73 + \left(55 a + 39\right)\cdot 73^{2} + \left(47 a + 9\right)\cdot 73^{3} + \left(50 a + 36\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $3 a + 57 + \left(66 a + 3\right)\cdot 73 + \left(17 a + 52\right)\cdot 73^{2} + \left(25 a + 24\right)\cdot 73^{3} + \left(22 a + 67\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $16 + 31\cdot 73 + 18\cdot 73^{2} + 5\cdot 73^{3} + 20\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $58 + 30\cdot 73 + 24\cdot 73^{2} + 72\cdot 73^{3} + 60\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 6 }$ $=$ $61 a + 29 + \left(56 a + 68\right)\cdot 73 + \left(28 a + 63\right)\cdot 73^{2} + \left(38 a + 46\right)\cdot 73^{3} + 35\cdot 73^{4} +O\left(73^{ 5 }\right)$

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