# Properties

 Label 4.820125.6t10.d.a Dimension $4$ Group $C_3^2:C_4$ Conductor $820125$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:C_4$ Conductor: $$820125$$$$\medspace = 3^{8} \cdot 5^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.4100625.2 Galois orbit size: $1$ Smallest permutation container: $C_3^2:C_4$ Parity: even Determinant: 1.5.2t1.a.a Projective image: $C_3:S_3.C_2$ Projective field: Galois closure of 6.2.4100625.2

## Defining polynomial

 $f(x)$ $=$ $x^{6} + 6 x^{4} - 3 x^{3} + 9 x^{2} - 9 x + 1$.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 13.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$

Roots:
 $r_{ 1 }$ $=$ $16 a + 8 + \left(6 a + 15\right)\cdot 19 + 18\cdot 19^{2} + \left(14 a + 6\right)\cdot 19^{3} + \left(13 a + 18\right)\cdot 19^{4} + \left(16 a + 2\right)\cdot 19^{5} + \left(18 a + 7\right)\cdot 19^{6} + \left(18 a + 17\right)\cdot 19^{7} + \left(13 a + 17\right)\cdot 19^{8} + \left(15 a + 4\right)\cdot 19^{9} + \left(3 a + 8\right)\cdot 19^{10} + \left(13 a + 4\right)\cdot 19^{11} + \left(8 a + 7\right)\cdot 19^{12} +O\left(19^{ 13 }\right)$ $r_{ 2 }$ $=$ $18 a + 4 + \left(7 a + 14\right)\cdot 19 + \left(15 a + 6\right)\cdot 19^{2} + \left(17 a + 11\right)\cdot 19^{3} + \left(11 a + 9\right)\cdot 19^{4} + \left(18 a + 13\right)\cdot 19^{5} + \left(18 a + 17\right)\cdot 19^{6} + \left(12 a + 11\right)\cdot 19^{7} + \left(14 a + 6\right)\cdot 19^{8} + \left(17 a + 12\right)\cdot 19^{9} + \left(17 a + 6\right)\cdot 19^{10} + \left(16 a + 17\right)\cdot 19^{11} + \left(16 a + 13\right)\cdot 19^{12} +O\left(19^{ 13 }\right)$ $r_{ 3 }$ $=$ $6 + 16\cdot 19 + 6\cdot 19^{2} + 10\cdot 19^{3} + 19^{4} + 10\cdot 19^{5} + 2\cdot 19^{6} + 3\cdot 19^{7} + 7\cdot 19^{8} + 7\cdot 19^{9} + 14\cdot 19^{10} + 9\cdot 19^{12} +O\left(19^{ 13 }\right)$ $r_{ 4 }$ $=$ $12 + 17\cdot 19^{2} + 12\cdot 19^{3} + 5\cdot 19^{4} + 4\cdot 19^{5} + 2\cdot 19^{6} + 19^{7} + 4\cdot 19^{8} + 10\cdot 19^{9} + 5\cdot 19^{10} + 4\cdot 19^{11} + 10\cdot 19^{12} +O\left(19^{ 13 }\right)$ $r_{ 5 }$ $=$ $3 a + 5 + \left(12 a + 6\right)\cdot 19 + \left(18 a + 12\right)\cdot 19^{2} + \left(4 a + 1\right)\cdot 19^{3} + \left(5 a + 18\right)\cdot 19^{4} + \left(2 a + 5\right)\cdot 19^{5} + 9\cdot 19^{6} + 17\cdot 19^{7} + \left(5 a + 12\right)\cdot 19^{8} + \left(3 a + 6\right)\cdot 19^{9} + \left(15 a + 15\right)\cdot 19^{10} + \left(5 a + 13\right)\cdot 19^{11} + \left(10 a + 2\right)\cdot 19^{12} +O\left(19^{ 13 }\right)$ $r_{ 6 }$ $=$ $a + 3 + \left(11 a + 4\right)\cdot 19 + \left(3 a + 14\right)\cdot 19^{2} + \left(a + 13\right)\cdot 19^{3} + \left(7 a + 3\right)\cdot 19^{4} + 19^{5} + 18\cdot 19^{6} + \left(6 a + 5\right)\cdot 19^{7} + \left(4 a + 8\right)\cdot 19^{8} + \left(a + 15\right)\cdot 19^{9} + \left(a + 6\right)\cdot 19^{10} + \left(2 a + 16\right)\cdot 19^{11} + \left(2 a + 13\right)\cdot 19^{12} +O\left(19^{ 13 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $9$ $2$ $(1,3)(2,4)$ $0$ $4$ $3$ $(1,3,5)$ $-2$ $4$ $3$ $(1,3,5)(2,4,6)$ $1$ $9$ $4$ $(1,4,3,2)(5,6)$ $0$ $9$ $4$ $(1,2,3,4)(5,6)$ $0$

The blue line marks the conjugacy class containing complex conjugation.