# Properties

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

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## 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 6.2.346112.1 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 $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$10 + 15\cdot 23 + 18\cdot 23^{2} + 10\cdot 23^{3} + 17\cdot 23^{4} + 23^{5} +O(23^{6})$$ 10 + 15*23 + 18*23^2 + 10*23^3 + 17*23^4 + 23^5+O(23^6) $r_{ 2 }$ $=$ $$16 + 19\cdot 23 + 19\cdot 23^{2} + 3\cdot 23^{3} + 8\cdot 23^{4} + 6\cdot 23^{5} +O(23^{6})$$ 16 + 19*23 + 19*23^2 + 3*23^3 + 8*23^4 + 6*23^5+O(23^6) $r_{ 3 }$ $=$ $$12 a + 18 + \left(4 a + 16\right)\cdot 23 + \left(21 a + 17\right)\cdot 23^{2} + \left(13 a + 2\right)\cdot 23^{3} + \left(6 a + 3\right)\cdot 23^{4} + \left(16 a + 9\right)\cdot 23^{5} +O(23^{6})$$ 12*a + 18 + (4*a + 16)*23 + (21*a + 17)*23^2 + (13*a + 2)*23^3 + (6*a + 3)*23^4 + (16*a + 9)*23^5+O(23^6) $r_{ 4 }$ $=$ $$11 a + 19 + \left(18 a + 13\right)\cdot 23 + \left(a + 9\right)\cdot 23^{2} + \left(9 a + 9\right)\cdot 23^{3} + \left(16 a + 2\right)\cdot 23^{4} + \left(6 a + 12\right)\cdot 23^{5} +O(23^{6})$$ 11*a + 19 + (18*a + 13)*23 + (a + 9)*23^2 + (9*a + 9)*23^3 + (16*a + 2)*23^4 + (6*a + 12)*23^5+O(23^6) $r_{ 5 }$ $=$ $$20 a + 7 + \left(14 a + 8\right)\cdot 23 + \left(15 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(19 a + 6\right)\cdot 23^{4} + \left(4 a + 13\right)\cdot 23^{5} +O(23^{6})$$ 20*a + 7 + (14*a + 8)*23 + (15*a + 16)*23^2 + (13*a + 3)*23^3 + (19*a + 6)*23^4 + (4*a + 13)*23^5+O(23^6) $r_{ 6 }$ $=$ $$3 a + 1 + \left(8 a + 18\right)\cdot 23 + \left(7 a + 9\right)\cdot 23^{2} + \left(9 a + 15\right)\cdot 23^{3} + \left(3 a + 8\right)\cdot 23^{4} + \left(18 a + 3\right)\cdot 23^{5} +O(23^{6})$$ 3*a + 1 + (8*a + 18)*23 + (7*a + 9)*23^2 + (9*a + 15)*23^3 + (3*a + 8)*23^4 + (18*a + 3)*23^5+O(23^6)

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

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

### Character values on conjugacy classes

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