# Properties

 Label 4.24128.6t13.a.a Dimension 4 Group $C_3^2:D_4$ Conductor $2^{6} \cdot 13 \cdot 29$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $24128= 2^{6} \cdot 13 \cdot 29$ Artin number field: Splitting field of 6.0.96512.1 defined by $f= x^{6} + x^{4} - 2 x^{3} - x^{2} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.377.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.0.96512.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $x^{2} + 49 x + 2$
Roots:
 $r_{ 1 }$ $=$ $16 + 13\cdot 53^{2} + 25\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $28 a + 4 + \left(35 a + 44\right)\cdot 53 + \left(40 a + 40\right)\cdot 53^{2} + \left(44 a + 52\right)\cdot 53^{3} + \left(11 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 a + 4 + \left(33 a + 48\right)\cdot 53 + \left(21 a + 2\right)\cdot 53^{2} + \left(14 a + 18\right)\cdot 53^{3} + \left(37 a + 38\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $25 a + 10 + \left(17 a + 52\right)\cdot 53 + \left(12 a + 8\right)\cdot 53^{2} + \left(8 a + 32\right)\cdot 53^{3} + \left(41 a + 19\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 5 }$ $=$ $40 + 52\cdot 53 + 36\cdot 53^{2} + 29\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 6 }$ $=$ $46 a + 32 + \left(19 a + 14\right)\cdot 53 + \left(31 a + 3\right)\cdot 53^{2} + \left(38 a + 1\right)\cdot 53^{3} + \left(15 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$

### 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.