# Properties

 Label 10.24002626001540481.30t164.a Dimension 10 Group $S_6$ Conductor $3^{12} \cdot 461^{4}$ Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $24002626001540481= 3^{12} \cdot 461^{4}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 30T164 Parity: Even Projective image: $S_6$ Projective field: Galois closure of 6.2.336069.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $x^{2} + 126 x + 3$
Roots:
 $r_{ 1 }$ $=$ $97 a + 77 + \left(40 a + 81\right)\cdot 127 + \left(86 a + 81\right)\cdot 127^{2} + \left(123 a + 68\right)\cdot 127^{3} + \left(4 a + 66\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 2 }$ $=$ $114 a + 53 + \left(45 a + 117\right)\cdot 127 + \left(66 a + 125\right)\cdot 127^{2} + \left(5 a + 93\right)\cdot 127^{3} + \left(20 a + 16\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 3 }$ $=$ $25 a + 71 + \left(27 a + 52\right)\cdot 127 + \left(120 a + 30\right)\cdot 127^{2} + \left(68 a + 65\right)\cdot 127^{3} + \left(74 a + 29\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 4 }$ $=$ $13 a + 40 + \left(81 a + 49\right)\cdot 127 + \left(60 a + 19\right)\cdot 127^{2} + \left(121 a + 33\right)\cdot 127^{3} + \left(106 a + 31\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 5 }$ $=$ $102 a + 96 + \left(99 a + 54\right)\cdot 127 + \left(6 a + 123\right)\cdot 127^{2} + \left(58 a + 13\right)\cdot 127^{3} + \left(52 a + 35\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$ $r_{ 6 }$ $=$ $30 a + 47 + \left(86 a + 25\right)\cdot 127 + 40 a\cdot 127^{2} + \left(3 a + 106\right)\cdot 127^{3} + \left(122 a + 74\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$

### 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 $c1$ $1$ $1$ $()$ $10$ $15$ $2$ $(1,2)(3,4)(5,6)$ $-2$ $15$ $2$ $(1,2)$ $2$ $45$ $2$ $(1,2)(3,4)$ $-2$ $40$ $3$ $(1,2,3)(4,5,6)$ $1$ $40$ $3$ $(1,2,3)$ $1$ $90$ $4$ $(1,2,3,4)(5,6)$ $0$ $90$ $4$ $(1,2,3,4)$ $0$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $1$ $120$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.