# Properties

 Label 10.595...176.30t164.a Dimension $10$ Group $S_6$ Conductor $5.958\times 10^{29}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $$595\!\cdots\!176$$$$\medspace = 2^{20} \cdot 19^{6} \cdot 479^{6}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.4.582464.1 Galois orbit size: $1$ Smallest permutation container: 30T164 Parity: even Projective image: $S_6$ Projective field: 6.4.582464.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6 x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$a + 1 + \left(3 a + 4\right)\cdot 7 + \left(6 a + 4\right)\cdot 7^{2} + \left(3 a + 5\right)\cdot 7^{3} + \left(4 a + 5\right)\cdot 7^{4} +O(7^{5})$$ $r_{ 2 }$ $=$ $$3 a + 5 + \left(2 a + 1\right)\cdot 7 + \left(5 a + 5\right)\cdot 7^{2} + \left(2 a + 6\right)\cdot 7^{3} + \left(a + 3\right)\cdot 7^{4} +O(7^{5})$$ $r_{ 3 }$ $=$ $$6 a + 4 + 6\cdot 7 + \left(6 a + 1\right)\cdot 7^{2} + \left(4 a + 1\right)\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} +O(7^{5})$$ $r_{ 4 }$ $=$ $$4 a + 1 + \left(4 a + 1\right)\cdot 7 + \left(a + 1\right)\cdot 7^{2} + \left(4 a + 4\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} +O(7^{5})$$ $r_{ 5 }$ $=$ $$6 a + 2 + \left(3 a + 6\right)\cdot 7 + \left(3 a + 3\right)\cdot 7^{3} + \left(2 a + 6\right)\cdot 7^{4} +O(7^{5})$$ $r_{ 6 }$ $=$ $$a + 3 + \left(6 a + 1\right)\cdot 7 + 2 a\cdot 7^{3} + 4 a\cdot 7^{4} +O(7^{5})$$

### 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 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.