# Properties

 Label 10.15813440003753001.30t164.a.a Dimension 10 Group $S_6$ Conductor $3^{6} \cdot 167^{6}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $15813440003753001= 3^{6} \cdot 167^{6}$ Artin number field: Splitting field of 6.0.83667.1 defined by $f= x^{6} + 2 x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 30T164 Parity: Even Determinant: 1.1.1t1.a.a Projective image: $S_6$ Projective field: Galois closure of 6.0.83667.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $x^{2} + 42 x + 3$
Roots:
 $r_{ 1 }$ $=$ $36 a + 14 + \left(6 a + 36\right)\cdot 43 + \left(23 a + 4\right)\cdot 43^{2} + \left(34 a + 40\right)\cdot 43^{3} + \left(6 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $32 + 25\cdot 43 + 37\cdot 43^{2} + 12\cdot 43^{3} + 19\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 a + 7 + \left(36 a + 7\right)\cdot 43 + \left(19 a + 21\right)\cdot 43^{2} + \left(8 a + 8\right)\cdot 43^{3} + \left(36 a + 31\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $6 a + 3 + \left(6 a + 25\right)\cdot 43 + \left(8 a + 34\right)\cdot 43^{2} + \left(41 a + 9\right)\cdot 43^{3} + \left(22 a + 14\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 5 }$ $=$ $21 + 9\cdot 43 + 37\cdot 43^{2} + 14\cdot 43^{3} + 9\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 6 }$ $=$ $37 a + 9 + \left(36 a + 25\right)\cdot 43 + \left(34 a + 36\right)\cdot 43^{2} + \left(a + 42\right)\cdot 43^{3} + \left(20 a + 38\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

### 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 $1$ $1$ $()$ $10$ $15$ $2$ $(1,2)(3,4)(5,6)$ $2$ $15$ $2$ $(1,2)$ $-2$ $45$ $2$ $(1,2)(3,4)$ $-2$ $40$ $3$ $(1,2,3)(4,5,6)$ $1$ $40$ $3$ $(1,2,3)$ $1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $90$ $4$ $(1,2,3,4)$ $0$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $-1$ $120$ $6$ $(1,2,3)(4,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.