# Properties

 Label 2.2352.6t3.a Dimension $2$ Group $D_{6}$ Conductor $2352$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2352$$$$\medspace = 2^{4} \cdot 3 \cdot 7^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.5531904.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: 3.1.588.1

## Galois action

### Roots of defining polynomial

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

### 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.