# Properties

 Label 10.2e8_3e20_5e4.30t176.2c1 Dimension 10 Group $S_6$ Conductor $2^{8} \cdot 3^{20} \cdot 5^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $557885504160000= 2^{8} \cdot 3^{20} \cdot 5^{4}$ Artin number field: Splitting field of $f= x^{6} + 3 x^{4} - 4 x^{3} + 9 x^{2} - 6 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 30T176 Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: $x^{2} + 145 x + 2$
Roots:
 $r_{ 1 }$ $=$ $102 a + 4 + \left(126 a + 81\right)\cdot 149 + \left(113 a + 123\right)\cdot 149^{2} + \left(88 a + 112\right)\cdot 149^{3} + \left(125 a + 8\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 2 }$ $=$ $47 a + 114 + \left(22 a + 38\right)\cdot 149 + \left(35 a + 5\right)\cdot 149^{2} + \left(60 a + 56\right)\cdot 149^{3} + \left(23 a + 124\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 3 }$ $=$ $87 a + 64 + \left(103 a + 96\right)\cdot 149 + \left(57 a + 98\right)\cdot 149^{2} + \left(62 a + 136\right)\cdot 149^{3} + \left(125 a + 85\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 4 }$ $=$ $62 a + 114 + \left(45 a + 125\right)\cdot 149 + \left(91 a + 76\right)\cdot 149^{2} + \left(86 a + 30\right)\cdot 149^{3} + \left(23 a + 78\right)\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 5 }$ $=$ $104 + 57\cdot 149 + 6\cdot 149^{2} + 90\cdot 149^{3} + 85\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 6 }$ $=$ $47 + 47\cdot 149 + 136\cdot 149^{2} + 20\cdot 149^{3} + 64\cdot 149^{4} +O\left(149^{ 5 }\right)$

### 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.