# Properties

 Label 2.980.6t3.e.a Dimension $2$ Group $D_{6}$ Conductor $980$ Root number $1$ 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 stem field: 6.0.3841600.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Determinant: 1.20.2t1.a.a Projective image: $S_3$ Projective stem field: 3.1.980.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2 x^{5} + 2 x^{4} - 10 x^{3} + 9 x^{2} + 12 x + 8$$  .

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:
 $r_{ 1 }$ $=$ $$11 a + 9 + \left(5 a + 12\right)\cdot 13 + 4\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + a\cdot 13^{4} + \left(4 a + 12\right)\cdot 13^{5} + \left(a + 2\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 2 }$ $=$ $$2 a + 7 + \left(7 a + 7\right)\cdot 13 + \left(12 a + 12\right)\cdot 13^{2} + \left(a + 2\right)\cdot 13^{3} + \left(11 a + 4\right)\cdot 13^{4} + \left(8 a + 1\right)\cdot 13^{5} + 11 a\cdot 13^{6} +O(13^{7})$$ $r_{ 3 }$ $=$ $$3 + 11\cdot 13 + 9\cdot 13^{2} + 4\cdot 13^{3} + 5\cdot 13^{5} + 11\cdot 13^{6} +O(13^{7})$$ $r_{ 4 }$ $=$ $$10 a + 7 + \left(6 a + 8\right)\cdot 13 + \left(9 a + 6\right)\cdot 13^{2} + a\cdot 13^{3} + \left(10 a + 12\right)\cdot 13^{4} + \left(4 a + 5\right)\cdot 13^{5} + \left(6 a + 9\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 5 }$ $=$ $$3 a + 4 + \left(6 a + 5\right)\cdot 13 + \left(3 a + 9\right)\cdot 13^{2} + \left(11 a + 5\right)\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} + 8 a\cdot 13^{5} + \left(6 a + 11\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 6 }$ $=$ $$11 + 6\cdot 13 + 8\cdot 13^{2} + 6\cdot 13^{3} + 13^{4} + 13^{5} + 4\cdot 13^{6} +O(13^{7})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,5)(3,6)$ $-2$ $3$ $2$ $(2,3)(5,6)$ $0$ $3$ $2$ $(1,4)(2,6)(3,5)$ $0$ $2$ $3$ $(1,2,3)(4,5,6)$ $-1$ $2$ $6$ $(1,5,3,4,2,6)$ $1$

The blue line marks the conjugacy class containing complex conjugation.