# Properties

 Label 4.20044127225.12t34.a.a Dimension 4 Group $C_3^2:D_4$ Conductor $5^{2} \cdot 929^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $20044127225= 5^{2} \cdot 929^{3}$ Artin number field: Splitting field of 6.2.116125.1 defined by $f= x^{6} - x^{5} + 4 x^{4} - x^{3} + 3 x^{2} - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 12T34 Parity: Even Determinant: 1.929.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.116125.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{2} + 7 x + 2$
Roots:
 $r_{ 1 }$ $=$ $2 + 10\cdot 11 + 5\cdot 11^{2} + 6\cdot 11^{3} + 10\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $7 a + 4 + \left(8 a + 2\right)\cdot 11 + \left(6 a + 2\right)\cdot 11^{2} + \left(2 a + 5\right)\cdot 11^{3} + \left(4 a + 10\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $5 + 8\cdot 11 + 9\cdot 11^{2} + 4\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 4 }$ $=$ $4 a + 4 + \left(5 a + 4\right)\cdot 11 + \left(a + 2\right)\cdot 11^{2} + 2 a\cdot 11^{3} + 10 a\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 5 }$ $=$ $7 a + 9 + \left(5 a + 10\right)\cdot 11 + \left(9 a + 2\right)\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + 5\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 6 }$ $=$ $4 a + 10 + \left(2 a + 7\right)\cdot 11 + \left(4 a + 9\right)\cdot 11^{2} + \left(8 a + 8\right)\cdot 11^{3} + \left(6 a + 2\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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