# Properties

 Label 10.705...664.30t164.b.a Dimension $10$ Group $S_6$ Conductor $7.051\times 10^{16}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $$70506920137457664$$$$\medspace = 2^{24} \cdot 3^{6} \cdot 7^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.2.3687936.1 Galois orbit size: $1$ Smallest permutation container: 30T164 Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_6$ Projective stem field: Galois closure of 6.2.3687936.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 4x^{4} - 2x^{3} + x^{2} - 2x - 5$$ x^6 - 4*x^4 - 2*x^3 + x^2 - 2*x - 5 .

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $$x^{2} + 192x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$167 a + 52 + \left(186 a + 78\right)\cdot 193 + \left(190 a + 25\right)\cdot 193^{2} + \left(157 a + 90\right)\cdot 193^{3} + \left(112 a + 99\right)\cdot 193^{4} +O(193^{5})$$ 167*a + 52 + (186*a + 78)*193 + (190*a + 25)*193^2 + (157*a + 90)*193^3 + (112*a + 99)*193^4+O(193^5) $r_{ 2 }$ $=$ $$49 + 23\cdot 193 + 35\cdot 193^{2} + 151\cdot 193^{3} + 93\cdot 193^{4} +O(193^{5})$$ 49 + 23*193 + 35*193^2 + 151*193^3 + 93*193^4+O(193^5) $r_{ 3 }$ $=$ $$26 a + 26 + \left(6 a + 98\right)\cdot 193 + \left(2 a + 29\right)\cdot 193^{2} + \left(35 a + 57\right)\cdot 193^{3} + \left(80 a + 54\right)\cdot 193^{4} +O(193^{5})$$ 26*a + 26 + (6*a + 98)*193 + (2*a + 29)*193^2 + (35*a + 57)*193^3 + (80*a + 54)*193^4+O(193^5) $r_{ 4 }$ $=$ $$62 + 104\cdot 193 + 147\cdot 193^{2} + 13\cdot 193^{3} + 32\cdot 193^{4} +O(193^{5})$$ 62 + 104*193 + 147*193^2 + 13*193^3 + 32*193^4+O(193^5) $r_{ 5 }$ $=$ $$134 a + 128 + \left(57 a + 175\right)\cdot 193 + \left(141 a + 128\right)\cdot 193^{2} + \left(69 a + 72\right)\cdot 193^{3} + \left(149 a + 13\right)\cdot 193^{4} +O(193^{5})$$ 134*a + 128 + (57*a + 175)*193 + (141*a + 128)*193^2 + (69*a + 72)*193^3 + (149*a + 13)*193^4+O(193^5) $r_{ 6 }$ $=$ $$59 a + 69 + \left(135 a + 99\right)\cdot 193 + \left(51 a + 19\right)\cdot 193^{2} + \left(123 a + 1\right)\cdot 193^{3} + \left(43 a + 93\right)\cdot 193^{4} +O(193^{5})$$ 59*a + 69 + (135*a + 99)*193 + (51*a + 19)*193^2 + (123*a + 1)*193^3 + (43*a + 93)*193^4+O(193^5)

## 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.