# Properties

 Label 4.20975.6t13.a Dimension $4$ Group $C_3^2:D_4$ Conductor $20975$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$20975$$$$\medspace = 5^{2} \cdot 839$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.4.104875.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: odd Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.4.104875.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $4$ $3$ $(1,4,6)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.