# Properties

 Label 10.43063e6.30t176.1c1 Dimension 10 Group $S_6$ Conductor $43063^{6}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $6377136176818436064486596209= 43063^{6}$ Artin number field: Splitting field of $f= x^{6} - x^{3} + x^{2} + 1$ 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 $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $x^{2} + 82 x + 3$
Roots:
 $r_{ 1 }$ $=$ $3 + 61\cdot 89 + 17\cdot 89^{2} + 6\cdot 89^{3} + 40\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 2 }$ $=$ $71 a + 24 + \left(26 a + 33\right)\cdot 89 + \left(82 a + 59\right)\cdot 89^{2} + \left(49 a + 16\right)\cdot 89^{3} + \left(11 a + 75\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 3 }$ $=$ $44 + 72\cdot 89 + 53\cdot 89^{2} + 30\cdot 89^{3} + 2\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 4 }$ $=$ $71 + 33\cdot 89 + 27\cdot 89^{2} + 7\cdot 89^{3} + 79\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 5 }$ $=$ $49 + 5\cdot 89 + 34\cdot 89^{2} + 11\cdot 89^{3} + 53\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 6 }$ $=$ $18 a + 76 + \left(62 a + 60\right)\cdot 89 + \left(6 a + 74\right)\cdot 89^{2} + \left(39 a + 16\right)\cdot 89^{3} + \left(77 a + 17\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$

### 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.