# Properties

 Label 10.33364831591822329.30t164.c.a Dimension 10 Group $S_6$ Conductor $3^{14} \cdot 17^{8}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $33364831591822329= 3^{14} \cdot 17^{8}$ Artin number field: Splitting field of 6.0.20295603.1 defined by $f= x^{6} - x^{5} - 2 x^{4} + x^{3} + 14 x^{2} - 17 x + 6$ 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 6.0.20295603.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $x^{2} + 192 x + 5$
Roots:
 $r_{ 1 }$ $=$ $67 + 139\cdot 193 + 134\cdot 193^{2} + 101\cdot 193^{3} + 74\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 2 }$ $=$ $98 + 189\cdot 193 + 100\cdot 193^{2} + 118\cdot 193^{3} + 21\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 3 }$ $=$ $176 a + 155 + \left(141 a + 115\right)\cdot 193 + \left(138 a + 43\right)\cdot 193^{2} + \left(172 a + 146\right)\cdot 193^{3} + \left(105 a + 112\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 4 }$ $=$ $98 a + 12 + \left(186 a + 175\right)\cdot 193 + \left(140 a + 55\right)\cdot 193^{2} + \left(98 a + 37\right)\cdot 193^{3} + \left(59 a + 85\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 5 }$ $=$ $95 a + 110 + \left(6 a + 70\right)\cdot 193 + \left(52 a + 10\right)\cdot 193^{2} + \left(94 a + 188\right)\cdot 193^{3} + \left(133 a + 45\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 6 }$ $=$ $17 a + 138 + \left(51 a + 81\right)\cdot 193 + \left(54 a + 40\right)\cdot 193^{2} + \left(20 a + 180\right)\cdot 193^{3} + \left(87 a + 45\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$

### 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.