# Properties

 Label 4.20975.6t13.a.a Dimension 4 Group $C_3^2:D_4$ Conductor $5^{2} \cdot 839$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $20975= 5^{2} \cdot 839$ Artin number field: Splitting field of 6.4.104875.1 defined by $f= x^{6} - 2 x^{5} - x^{4} + x^{3} + 2 x^{2} + x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Odd Determinant: 1.839.2t1.a.a Projective image: $S_3\wr C_2$ 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 $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $16 + 11\cdot 19 + 15\cdot 19^{2} + 13\cdot 19^{3} + 9\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 a + 4 + \left(7 a + 2\right)\cdot 19 + \left(17 a + 6\right)\cdot 19^{2} + \left(9 a + 3\right)\cdot 19^{3} + \left(11 a + 16\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 a + 13 + 11 a\cdot 19 + \left(a + 16\right)\cdot 19^{2} + \left(9 a + 14\right)\cdot 19^{3} + \left(7 a + 17\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $17 a + 3 + \left(16 a + 13\right)\cdot 19 + 18 a\cdot 19^{2} + \left(13 a + 5\right)\cdot 19^{3} + \left(16 a + 3\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $3 + 16\cdot 19 + 15\cdot 19^{2} + 4\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $2 a + 1 + \left(2 a + 13\right)\cdot 19 + 2\cdot 19^{2} + 5 a\cdot 19^{3} + \left(2 a + 6\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

### 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 value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $4$ $3$ $(1,4,6)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.