# Properties

 Label 2.416.6t3.c Dimension $2$ Group $D_{6}$ Conductor $416$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$416$$$$\medspace = 2^{5} \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.173056.2 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.104.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots:
 $r_{ 1 }$ $=$ $8 a + 4 + \left(16 a + 4\right)\cdot 29 + \left(a + 17\right)\cdot 29^{2} + \left(4 a + 22\right)\cdot 29^{3} + 13 a\cdot 29^{4} + \left(14 a + 3\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 2 }$ $=$ $21 a + 15 + \left(12 a + 19\right)\cdot 29 + \left(27 a + 8\right)\cdot 29^{2} + \left(24 a + 12\right)\cdot 29^{3} + \left(15 a + 4\right)\cdot 29^{4} + \left(14 a + 4\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 3 }$ $=$ $19 + 23\cdot 29 + 25\cdot 29^{2} + 5\cdot 29^{3} + 5\cdot 29^{4} + 7\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 4 }$ $=$ $21 a + 25 + \left(12 a + 24\right)\cdot 29 + \left(27 a + 11\right)\cdot 29^{2} + \left(24 a + 6\right)\cdot 29^{3} + \left(15 a + 28\right)\cdot 29^{4} + \left(14 a + 25\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 5 }$ $=$ $8 a + 14 + \left(16 a + 9\right)\cdot 29 + \left(a + 20\right)\cdot 29^{2} + \left(4 a + 16\right)\cdot 29^{3} + \left(13 a + 24\right)\cdot 29^{4} + \left(14 a + 24\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 6 }$ $=$ $10 + 5\cdot 29 + 3\cdot 29^{2} + 23\cdot 29^{3} + 23\cdot 29^{4} + 21\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

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

### Character values on conjugacy classes

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