# Properties

 Label 4.3e3_907.6t13.2c1 Dimension 4 Group $C_3^2:D_4$ Conductor $3^{3} \cdot 907$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $24489= 3^{3} \cdot 907$ Artin number field: Splitting field of $f= x^{6} - x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 3 x + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.3_907.2t1.1c1

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

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

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

### Character values on conjugacy classes

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