# Properties

 Label 5.425473129.12t183.a.a Dimension $5$ Group $S_6$ Conductor $425473129$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $$425473129$$$$\medspace = 20627^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 6.0.20627.1 Galois orbit size: $1$ Smallest permutation container: 12T183 Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_6$ Projective stem field: 6.0.20627.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + x^{4} - 2 x^{3} + 2 x^{2} - x + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $$x^{2} + 274 x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$62 a + 226 + \left(43 a + 116\right)\cdot 277 + \left(152 a + 9\right)\cdot 277^{2} + \left(231 a + 84\right)\cdot 277^{3} + \left(175 a + 85\right)\cdot 277^{4} +O(277^{5})$$ $r_{ 2 }$ $=$ $$230 + 135\cdot 277 + 119\cdot 277^{2} + 146\cdot 277^{3} + 55\cdot 277^{4} +O(277^{5})$$ $r_{ 3 }$ $=$ $$118 + 184\cdot 277 + 194\cdot 277^{2} + 65\cdot 277^{3} + 153\cdot 277^{4} +O(277^{5})$$ $r_{ 4 }$ $=$ $$41 + 146\cdot 277 + 30\cdot 277^{2} + 9\cdot 277^{3} + 24\cdot 277^{4} +O(277^{5})$$ $r_{ 5 }$ $=$ $$215 a + 135 + \left(233 a + 184\right)\cdot 277 + \left(124 a + 145\right)\cdot 277^{2} + \left(45 a + 72\right)\cdot 277^{3} + \left(101 a + 104\right)\cdot 277^{4} +O(277^{5})$$ $r_{ 6 }$ $=$ $$82 + 63\cdot 277 + 54\cdot 277^{2} + 176\cdot 277^{3} + 131\cdot 277^{4} +O(277^{5})$$

## 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 $1$ $1$ $()$ $5$ $15$ $2$ $(1,2)(3,4)(5,6)$ $-3$ $15$ $2$ $(1,2)$ $1$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $2$ $40$ $3$ $(1,2,3)$ $-1$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $90$ $4$ $(1,2,3,4)$ $-1$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $0$ $120$ $6$ $(1,2,3)(4,5)$ $1$

The blue line marks the conjugacy class containing complex conjugation.