# Properties

 Label 4.6793.5t5.a.a Dimension 4 Group $S_5$ Conductor $6793$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $6793$ Artin number field: Splitting field of 5.1.6793.1 defined by $f= x^{5} - x^{4} + x^{3} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.6793.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.1.6793.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $39 + 262\cdot 431 + 185\cdot 431^{2} + 249\cdot 431^{3} + 191\cdot 431^{4} +O\left(431^{ 5 }\right)$ $r_{ 2 }$ $=$ $108 + 63\cdot 431 + 206\cdot 431^{2} + 48\cdot 431^{3} + 100\cdot 431^{4} +O\left(431^{ 5 }\right)$ $r_{ 3 }$ $=$ $132 + 200\cdot 431 + 311\cdot 431^{2} + 170\cdot 431^{3} + 275\cdot 431^{4} +O\left(431^{ 5 }\right)$ $r_{ 4 }$ $=$ $158 + 353\cdot 431 + 323\cdot 431^{2} + 179\cdot 431^{3} + 201\cdot 431^{4} +O\left(431^{ 5 }\right)$ $r_{ 5 }$ $=$ $426 + 413\cdot 431 + 265\cdot 431^{2} + 213\cdot 431^{3} + 93\cdot 431^{4} +O\left(431^{ 5 }\right)$

### 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.