# Properties

 Label 2.1575.6t3.b Dimension $2$ Group $D_{6}$ Conductor $1575$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$1575$$$$\medspace = 3^{2} \cdot 5^{2} \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.4134375.1 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 $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{2} + 16 x + 3$
Roots:
 $r_{ 1 }$ $=$ $15 a + 10 + \left(2 a + 7\right)\cdot 17 + \left(2 a + 8\right)\cdot 17^{2} + \left(5 a + 13\right)\cdot 17^{3} + \left(9 a + 1\right)\cdot 17^{4} + \left(7 a + 8\right)\cdot 17^{5} + 3 a\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 2 }$ $=$ $5 + 3\cdot 17 + 2\cdot 17^{2} + 15\cdot 17^{3} + 7\cdot 17^{4} + 7\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 3 }$ $=$ $7 a + 16 + \left(8 a + 16\right)\cdot 17 + \left(15 a + 16\right)\cdot 17^{2} + \left(7 a + 2\right)\cdot 17^{3} + \left(10 a + 12\right)\cdot 17^{4} + \left(9 a + 14\right)\cdot 17^{5} + 4\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 4 }$ $=$ $2 a + 8 + \left(14 a + 12\right)\cdot 17 + \left(14 a + 7\right)\cdot 17^{2} + \left(11 a + 16\right)\cdot 17^{3} + \left(7 a + 5\right)\cdot 17^{4} + \left(9 a + 6\right)\cdot 17^{5} + \left(13 a + 13\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 5 }$ $=$ $7 + 9\cdot 17 + 8\cdot 17^{2} + 7\cdot 17^{3} + 8\cdot 17^{4} + 7\cdot 17^{5} + 12\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 6 }$ $=$ $10 a + 6 + \left(8 a + 1\right)\cdot 17 + \left(a + 7\right)\cdot 17^{2} + \left(9 a + 12\right)\cdot 17^{3} + \left(6 a + 14\right)\cdot 17^{4} + \left(7 a + 13\right)\cdot 17^{5} + \left(16 a + 12\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$

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

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

### Character values on conjugacy classes

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