# Properties

 Label 4.3897.6t13.b Dimension $4$ Group $C_3^2:D_4$ Conductor $3897$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$3897$$$$\medspace = 3^{2} \cdot 433$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.11691.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.0.11691.1

## 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 $\mathbb{Q}_{ 13 }$: $$x^{2} + 12x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$10 a + 11 + \left(4 a + 7\right)\cdot 13 + \left(4 a + 3\right)\cdot 13^{2} + \left(8 a + 4\right)\cdot 13^{3} + \left(10 a + 11\right)\cdot 13^{4} +O(13^{5})$$ 10*a + 11 + (4*a + 7)*13 + (4*a + 3)*13^2 + (8*a + 4)*13^3 + (10*a + 11)*13^4+O(13^5) $r_{ 2 }$ $=$ $$a + 3 + 9\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + \left(4 a + 9\right)\cdot 13^{3} + \left(7 a + 2\right)\cdot 13^{4} +O(13^{5})$$ a + 3 + 9*13 + (10*a + 9)*13^2 + (4*a + 9)*13^3 + (7*a + 2)*13^4+O(13^5) $r_{ 3 }$ $=$ $$3 + 10\cdot 13 + 2\cdot 13^{2} + 2\cdot 13^{3} + 10\cdot 13^{4} +O(13^{5})$$ 3 + 10*13 + 2*13^2 + 2*13^3 + 10*13^4+O(13^5) $r_{ 4 }$ $=$ $$11 + 10\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})$$ 11 + 10*13^3 + 8*13^4+O(13^5) $r_{ 5 }$ $=$ $$12 a + 4 + \left(12 a + 8\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + \left(8 a + 4\right)\cdot 13^{3} + \left(5 a + 5\right)\cdot 13^{4} +O(13^{5})$$ 12*a + 4 + (12*a + 8)*13 + (2*a + 6)*13^2 + (8*a + 4)*13^3 + (5*a + 5)*13^4+O(13^5) $r_{ 6 }$ $=$ $$3 a + 8 + \left(8 a + 2\right)\cdot 13 + \left(8 a + 3\right)\cdot 13^{2} + \left(4 a + 8\right)\cdot 13^{3} + 2 a\cdot 13^{4} +O(13^{5})$$ 3*a + 8 + (8*a + 2)*13 + (8*a + 3)*13^2 + (4*a + 8)*13^3 + 2*a*13^4+O(13^5)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(3,5)$ $2$ $9$ $2$ $(3,5)(4,6)$ $0$ $4$ $3$ $(1,4,6)(2,3,5)$ $-2$ $4$ $3$ $(1,4,6)$ $1$ $18$ $4$ $(1,2)(3,6,5,4)$ $0$ $12$ $6$ $(1,3,4,5,6,2)$ $0$ $12$ $6$ $(1,4,6)(3,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.