# Properties

 Label 10.10000000000000000.30t164.a.a Dimension 10 Group $S_6$ Conductor $2^{16} \cdot 5^{16}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $10000000000000000= 2^{16} \cdot 5^{16}$ Artin number field: Splitting field of 6.2.12500000.1 defined by $f= x^{6} - 3 x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 4 x - 8$ 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 6.2.12500000.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots:
 $r_{ 1 }$ $=$ $45 a + 25 + \left(3 a + 23\right)\cdot 47 + \left(29 a + 13\right)\cdot 47^{2} + \left(7 a + 1\right)\cdot 47^{3} + \left(24 a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 + 8\cdot 47 + 45\cdot 47^{2} + 9\cdot 47^{3} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $2 a + 21 + \left(43 a + 33\right)\cdot 47 + \left(17 a + 20\right)\cdot 47^{2} + \left(39 a + 34\right)\cdot 47^{3} + \left(22 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $15 + 38\cdot 47 + 20\cdot 47^{2} + 41\cdot 47^{3} + 12\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $7 a + 7 + \left(18 a + 4\right)\cdot 47 + \left(39 a + 37\right)\cdot 47^{2} + \left(29 a + 16\right)\cdot 47^{3} + \left(44 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 6 }$ $=$ $40 a + 21 + \left(28 a + 33\right)\cdot 47 + \left(7 a + 3\right)\cdot 47^{2} + \left(17 a + 37\right)\cdot 47^{3} + \left(2 a + 24\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

### 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.