# Properties

 Label 4.5e2_409.6t13.1c1 Dimension 4 Group $C_3^2:D_4$ Conductor $5^{2} \cdot 409$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $10225= 5^{2} \cdot 409$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{2} + 3 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.409.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $x^{2} + 108 x + 6$
Roots:
 $r_{ 1 }$ $=$ $35 + 32\cdot 109 + 72\cdot 109^{2} + 55\cdot 109^{3} + 82\cdot 109^{4} +O\left(109^{ 5 }\right)$ $r_{ 2 }$ $=$ $98 a + 49 + \left(47 a + 68\right)\cdot 109 + \left(80 a + 78\right)\cdot 109^{2} + \left(48 a + 91\right)\cdot 109^{3} + \left(80 a + 73\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ $r_{ 3 }$ $=$ $71 a + 2 + \left(80 a + 88\right)\cdot 109 + \left(33 a + 41\right)\cdot 109^{2} + \left(74 a + 6\right)\cdot 109^{3} + \left(44 a + 28\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ $r_{ 4 }$ $=$ $23 + 22\cdot 109 + 28\cdot 109^{2} + 66\cdot 109^{3} + 38\cdot 109^{4} +O\left(109^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 38 + \left(61 a + 18\right)\cdot 109 + \left(28 a + 2\right)\cdot 109^{2} + \left(60 a + 60\right)\cdot 109^{3} + \left(28 a + 105\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$ $r_{ 6 }$ $=$ $38 a + 73 + \left(28 a + 97\right)\cdot 109 + \left(75 a + 103\right)\cdot 109^{2} + \left(34 a + 46\right)\cdot 109^{3} + \left(64 a + 107\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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