# Properties

 Label 5.3e8_7e4.6t15.2c1 Dimension 5 Group $A_6$ Conductor $3^{8} \cdot 7^{4}$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $5$ Group: $A_6$ Conductor: $15752961= 3^{8} \cdot 7^{4}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} + 9 x^{3} - 18 x^{2} - 9 x + 18$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_6$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $x^{2} + 149 x + 6$
Roots:
 $r_{ 1 }$ $=$ $3 a + 43 + \left(58 a + 148\right)\cdot 151 + \left(49 a + 52\right)\cdot 151^{2} + \left(129 a + 106\right)\cdot 151^{3} + \left(108 a + 148\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 2 }$ $=$ $48 a + 76 + \left(8 a + 37\right)\cdot 151 + \left(104 a + 23\right)\cdot 151^{2} + \left(108 a + 76\right)\cdot 151^{3} + \left(150 a + 133\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 3 }$ $=$ $148 a + 49 + \left(92 a + 110\right)\cdot 151 + \left(101 a + 93\right)\cdot 151^{2} + \left(21 a + 13\right)\cdot 151^{3} + \left(42 a + 86\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 4 }$ $=$ $123 + 50\cdot 151 + 86\cdot 151^{2} + 128\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 5 }$ $=$ $103 a + 21 + \left(142 a + 6\right)\cdot 151 + \left(46 a + 72\right)\cdot 151^{2} + \left(42 a + 38\right)\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 6 }$ $=$ $144 + 99\cdot 151 + 124\cdot 151^{2} + 89\cdot 151^{3} + 35\cdot 151^{4} +O\left(151^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $5$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $40$ $3$ $(1,2,3)$ $2$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $72$ $5$ $(1,2,3,4,5)$ $0$ $72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.