# Properties

 Label 4.3e5_43.6t13.2c1 Dimension 4 Group $C_3^2:D_4$ Conductor $3^{5} \cdot 43$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $10449= 3^{5} \cdot 43$ Artin number field: Splitting field of $f= x^{6} - 3 x^{4} - 2 x^{3} + 3 x^{2} + 3 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.3_43.2t1.1c1

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

### 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.