# Properties

 Label 2.980.6t3.b Dimension $2$ Group $D_{6}$ Conductor $980$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$980$$$$\medspace = 2^{2} \cdot 5 \cdot 7^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.26891200.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: 3.1.980.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$4 + 19 + 15\cdot 19^{2} + 5\cdot 19^{3} + 10\cdot 19^{4} + 10\cdot 19^{5} + 14\cdot 19^{6} +O(19^{7})$$ $r_{ 2 }$ $=$ $$8 + 10\cdot 19 + 18\cdot 19^{2} + 18\cdot 19^{3} + 5\cdot 19^{4} + 4\cdot 19^{5} + 4\cdot 19^{6} +O(19^{7})$$ $r_{ 3 }$ $=$ $$2 a + 7 + \left(3 a + 8\right)\cdot 19 + \left(2 a + 2\right)\cdot 19^{2} + 15 a\cdot 19^{3} + 2\cdot 19^{4} + \left(14 a + 7\right)\cdot 19^{5} + \left(18 a + 9\right)\cdot 19^{6} +O(19^{7})$$ $r_{ 4 }$ $=$ $$15 a + 8 + \left(2 a + 10\right)\cdot 19 + 2 a\cdot 19^{2} + \left(7 a + 7\right)\cdot 19^{3} + \left(4 a + 17\right)\cdot 19^{4} + \left(7 a + 5\right)\cdot 19^{5} + 2 a\cdot 19^{6} +O(19^{7})$$ $r_{ 5 }$ $=$ $$4 a + 4 + \left(16 a + 17\right)\cdot 19 + \left(16 a + 18\right)\cdot 19^{2} + \left(11 a + 11\right)\cdot 19^{3} + \left(14 a + 14\right)\cdot 19^{4} + \left(11 a + 8\right)\cdot 19^{5} + \left(16 a + 14\right)\cdot 19^{6} +O(19^{7})$$ $r_{ 6 }$ $=$ $$17 a + 9 + \left(15 a + 9\right)\cdot 19 + \left(16 a + 1\right)\cdot 19^{2} + \left(3 a + 13\right)\cdot 19^{3} + \left(18 a + 6\right)\cdot 19^{4} + \left(4 a + 1\right)\cdot 19^{5} + 14\cdot 19^{6} +O(19^{7})$$

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

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

### Character values on conjugacy classes

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