# Properties

 Label 4.15993.6t13.a.a Dimension 4 Group $C_3^2:D_4$ Conductor $3^{2} \cdot 1777$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $15993= 3^{2} \cdot 1777$ Artin number field: Splitting field of 6.0.47979.1 defined by $f= x^{6} - 3 x^{5} + 2 x^{4} - x^{3} + x^{2} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.1777.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.0.47979.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots:
 $r_{ 1 }$ $=$ $9 a + 11 + \left(4 a + 5\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(9 a + 9\right)\cdot 13^{3} + \left(8 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 7\cdot 13 + 8\cdot 13^{2} + 11\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $6 a + 1 + \left(12 a + 10\right)\cdot 13 + 12\cdot 13^{2} + \left(12 a + 1\right)\cdot 13^{3} + 9 a\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $4 a + 7 + \left(8 a + 1\right)\cdot 13 + \left(a + 2\right)\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(4 a + 3\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 5 }$ $=$ $7 a + 7 + 3\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 6 }$ $=$ $10 + 10\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 9\cdot 13^{4} +O\left(13^{ 5 }\right)$

### 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.