# Properties

 Label 4.16400.6t13.d.a Dimension 4 Group $C_3^2:D_4$ Conductor $2^{4} \cdot 5^{2} \cdot 41$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $16400= 2^{4} \cdot 5^{2} \cdot 41$ Artin number field: Splitting field of 6.0.65600.1 defined by $f= x^{6} - 2 x^{5} - x^{4} + 2 x^{3} + 2 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.41.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.0.65600.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots:
 $r_{ 1 }$ $=$ $19 a + 29 + \left(16 a + 5\right)\cdot 37 + \left(9 a + 36\right)\cdot 37^{2} + \left(7 a + 2\right)\cdot 37^{3} + \left(10 a + 36\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $23 a + 34 + \left(17 a + 24\right)\cdot 37 + \left(13 a + 35\right)\cdot 37^{2} + 17 a\cdot 37^{3} + \left(18 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $18 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(27 a + 20\right)\cdot 37^{2} + \left(29 a + 22\right)\cdot 37^{3} + \left(26 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $26 + 13\cdot 37 + 3\cdot 37^{2} + 16\cdot 37^{3} + 3\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $15 + 15\cdot 37 + 17\cdot 37^{2} + 11\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 6 }$ $=$ $14 a + 15 + \left(19 a + 35\right)\cdot 37 + \left(23 a + 34\right)\cdot 37^{2} + \left(19 a + 19\right)\cdot 37^{3} + \left(18 a + 26\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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