# Properties

 Label 2.175.6t3.a Dimension $2$ Group $D_{6}$ Conductor $175$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$175$$$$\medspace = 5^{2} \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.153125.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.175.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $15 a + 1 + \left(5 a + 1\right)\cdot 19 + \left(17 a + 13\right)\cdot 19^{2} + 13\cdot 19^{3} + \left(6 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 a + 2 + \left(13 a + 12\right)\cdot 19 + \left(16 a + 9\right)\cdot 19^{2} + \left(9 a + 5\right)\cdot 19^{3} + \left(12 a + 8\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 + 9\cdot 19 + 10\cdot 19^{2} + 3\cdot 19^{3} + 14\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $13 a + 8 + 5 a\cdot 19 + \left(2 a + 13\right)\cdot 19^{2} + \left(9 a + 17\right)\cdot 19^{3} + \left(6 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $4 a + 16 + \left(13 a + 10\right)\cdot 19 + \left(a + 5\right)\cdot 19^{2} + \left(18 a + 16\right)\cdot 19^{3} + \left(12 a + 8\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $5 + 4\cdot 19 + 5\cdot 19^{2} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(2,6)(3,5)$ $(1,2)(3,6)(4,5)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,5)(3,6)$ $-2$ $3$ $2$ $(1,2)(3,6)(4,5)$ $0$ $3$ $2$ $(1,3)(4,6)$ $0$ $2$ $3$ $(1,5,3)(2,6,4)$ $-1$ $2$ $6$ $(1,6,5,4,3,2)$ $1$
The blue line marks the conjugacy class containing complex conjugation.