# Properties

 Label 4.20025.6t13.a.a Dimension 4 Group $C_3^2:D_4$ Conductor $3^{2} \cdot 5^{2} \cdot 89$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $20025= 3^{2} \cdot 5^{2} \cdot 89$ Artin number field: Splitting field of 6.2.100125.1 defined by $f= x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} + x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.89.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.2.100125.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $x^{2} + 69 x + 7$
Roots:
 $r_{ 1 }$ $=$ $7 + 58\cdot 71 + 14\cdot 71^{2} + 19\cdot 71^{3} + 12\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 a + 52 + \left(48 a + 4\right)\cdot 71 + \left(32 a + 13\right)\cdot 71^{2} + \left(48 a + 17\right)\cdot 71^{3} + \left(60 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 3 }$ $=$ $27 a + 41 + \left(22 a + 68\right)\cdot 71 + \left(54 a + 55\right)\cdot 71^{2} + \left(15 a + 1\right)\cdot 71^{3} + \left(10 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 4 }$ $=$ $44 a + 24 + \left(48 a + 15\right)\cdot 71 + 16 a\cdot 71^{2} + \left(55 a + 50\right)\cdot 71^{3} + \left(60 a + 31\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 5 }$ $=$ $15 + 48\cdot 71 + 27\cdot 71^{2} + 43\cdot 71^{3} + 3\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 6 }$ $=$ $59 a + 5 + \left(22 a + 18\right)\cdot 71 + \left(38 a + 30\right)\cdot 71^{2} + \left(22 a + 10\right)\cdot 71^{3} + \left(10 a + 70\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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