# Properties

 Label 4.3e2_5e3_47e2.6t10.3 Dimension 4 Group $C_3^2:C_4$ Conductor $3^{2} \cdot 5^{3} \cdot 47^{2}$ Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:C_4$ Conductor: $2485125= 3^{2} \cdot 5^{3} \cdot 47^{2}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 9 x^{4} - 9 x^{3} + 9 x^{2} + 12 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:C_4$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots:
 $r_{ 1 }$ $=$ $16 + 21\cdot 29 + 2\cdot 29^{2} + 3\cdot 29^{3} + 9\cdot 29^{4} + 24\cdot 29^{5} + 14\cdot 29^{6} + 4\cdot 29^{7} + 5\cdot 29^{8} + 20\cdot 29^{9} + 29^{10} + 10\cdot 29^{11} + 7\cdot 29^{12} +O\left(29^{ 13 }\right)$ $r_{ 2 }$ $=$ $6 a + 10 + \left(18 a + 1\right)\cdot 29 + \left(27 a + 6\right)\cdot 29^{2} + \left(2 a + 19\right)\cdot 29^{3} + \left(9 a + 14\right)\cdot 29^{4} + \left(26 a + 1\right)\cdot 29^{5} + \left(3 a + 19\right)\cdot 29^{6} + \left(9 a + 11\right)\cdot 29^{7} + \left(13 a + 21\right)\cdot 29^{8} + \left(3 a + 25\right)\cdot 29^{9} + \left(13 a + 7\right)\cdot 29^{10} + \left(13 a + 15\right)\cdot 29^{11} + \left(26 a + 10\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$ $r_{ 3 }$ $=$ $23 a + 11 + \left(10 a + 28\right)\cdot 29 + \left(a + 9\right)\cdot 29^{2} + \left(26 a + 6\right)\cdot 29^{3} + \left(19 a + 28\right)\cdot 29^{4} + \left(2 a + 7\right)\cdot 29^{5} + \left(25 a + 12\right)\cdot 29^{6} + \left(19 a + 24\right)\cdot 29^{7} + \left(15 a + 20\right)\cdot 29^{8} + 25 a\cdot 29^{9} + \left(15 a + 12\right)\cdot 29^{10} + \left(15 a + 11\right)\cdot 29^{11} + \left(2 a + 13\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$ $r_{ 4 }$ $=$ $6 a + 4 + \left(27 a + 15\right)\cdot 29 + \left(13 a + 18\right)\cdot 29^{2} + \left(3 a + 11\right)\cdot 29^{3} + \left(10 a + 22\right)\cdot 29^{4} + \left(23 a + 20\right)\cdot 29^{5} + \left(4 a + 21\right)\cdot 29^{6} + \left(21 a + 19\right)\cdot 29^{7} + \left(20 a + 10\right)\cdot 29^{8} + \left(8 a + 17\right)\cdot 29^{9} + \left(2 a + 24\right)\cdot 29^{10} + \left(12 a + 26\right)\cdot 29^{11} + \left(12 a + 17\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$ $r_{ 5 }$ $=$ $23 a + 5 + a\cdot 29 + \left(15 a + 3\right)\cdot 29^{2} + \left(25 a + 15\right)\cdot 29^{3} + \left(18 a + 11\right)\cdot 29^{4} + \left(5 a + 11\right)\cdot 29^{5} + \left(24 a + 22\right)\cdot 29^{6} + \left(7 a + 4\right)\cdot 29^{7} + \left(8 a + 6\right)\cdot 29^{8} + \left(20 a + 11\right)\cdot 29^{9} + \left(26 a + 27\right)\cdot 29^{10} + \left(16 a + 26\right)\cdot 29^{11} + \left(16 a + 9\right)\cdot 29^{12} +O\left(29^{ 13 }\right)$ $r_{ 6 }$ $=$ $15 + 20\cdot 29 + 17\cdot 29^{2} + 2\cdot 29^{3} + 29^{4} + 21\cdot 29^{5} + 25\cdot 29^{6} + 21\cdot 29^{7} + 22\cdot 29^{8} + 11\cdot 29^{9} + 13\cdot 29^{10} + 25\cdot 29^{11} + 27\cdot 29^{12} +O\left(29^{ 13 }\right)$

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

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

### Character values on conjugacy classes

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