# Properties

 Label 5.236549e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $236549^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $13236200869377149= 236549^{3}$ Artin number field: Splitting field of $f= x^{5} - x^{4} - 6 x^{3} + 7 x^{2} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.236549.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 461 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 98 + 99\cdot 461 + 277\cdot 461^{2} + 443\cdot 461^{3} + 317\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 2 } &= 203 + 279\cdot 461 + 175\cdot 461^{2} + 315\cdot 461^{3} + 430\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 3 } &= 247 + 346\cdot 461 + 218\cdot 461^{2} + 382\cdot 461^{3} + 138\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 4 } &= 386 + 441\cdot 461 + 9\cdot 461^{2} + 144\cdot 461^{3} + 38\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 5 } &= 450 + 215\cdot 461 + 240\cdot 461^{2} + 97\cdot 461^{3} + 457\cdot 461^{4} +O\left(461^{ 5 }\right) \\ \end{aligned}

### 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 $1$ $1$ $()$ $5$ $10$ $2$ $(1,2)$ $-1$ $15$ $2$ $(1,2)(3,4)$ $1$ $20$ $3$ $(1,2,3)$ $-1$ $30$ $4$ $(1,2,3,4)$ $1$ $24$ $5$ $(1,2,3,4,5)$ $0$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.