# Properties

 Label 2.2736.6t3.a Dimension $2$ Group $D_{6}$ Conductor $2736$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2736$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.2495232.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: 3.1.76.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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