# Properties

 Label 2.300.6t3.a Dimension $2$ Group $D_{6}$ Conductor $300$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$300$$$$\medspace = 2^{2} \cdot 3 \cdot 5^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.450000.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: 3.1.300.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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