# Properties

 Label 5.2e4_47_61.6t16.1c1 Dimension 5 Group $S_6$ Conductor $2^{4} \cdot 47 \cdot 61$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $45872= 2^{4} \cdot 47 \cdot 61$ Artin number field: Splitting field of $f= x^{6} - x^{5} - x^{3} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_6$ Parity: Odd Determinant: 1.47_61.2t1.1c1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $5$ $15$ $2$ $(1,2)(3,4)(5,6)$ $-1$ $15$ $2$ $(1,2)$ $3$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $40$ $3$ $(1,2,3)$ $2$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $90$ $4$ $(1,2,3,4)$ $1$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $-1$ $120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.