# Properties

 Label 4.5653.5t5.a.a Dimension 4 Group $S_5$ Conductor $5653$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $5653$ Artin number field: Splitting field of 5.1.5653.1 defined by $f= x^{5} - x^{4} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.5653.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.1.5653.1

## 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:
 $r_{ 1 }$ $=$ $21 a + 20 + \left(10 a + 10\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(18 a + 17\right)\cdot 23^{3} + 8 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $17 + 2\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 21\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $2 a + 16 + \left(12 a + 11\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(4 a + 3\right)\cdot 23^{3} + 14 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 9 + \left(21 a + 17\right)\cdot 23 + \left(5 a + 14\right)\cdot 23^{2} + \left(19 a + 21\right)\cdot 23^{3} + \left(17 a + 14\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $12 a + 8 + \left(a + 3\right)\cdot 23 + \left(17 a + 5\right)\cdot 23^{2} + \left(3 a + 8\right)\cdot 23^{3} + \left(5 a + 8\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.