# Properties

 Label 4.4432.5t5.b.a Dimension $4$ Group $S_5$ Conductor $4432$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $$4432$$$$\medspace = 2^{4} \cdot 277$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 5.1.4432.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Determinant: 1.277.2t1.a.a Projective image: $S_5$ Projective stem field: Galois closure of 5.1.4432.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $$x^{2} + 45x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$12 + 13\cdot 47 + 46\cdot 47^{2} + 15\cdot 47^{3} + 18\cdot 47^{4} +O(47^{5})$$ 12 + 13*47 + 46*47^2 + 15*47^3 + 18*47^4+O(47^5) $r_{ 2 }$ $=$ $$45 a + 27 + \left(16 a + 7\right)\cdot 47 + \left(10 a + 18\right)\cdot 47^{2} + \left(43 a + 36\right)\cdot 47^{3} + \left(16 a + 14\right)\cdot 47^{4} +O(47^{5})$$ 45*a + 27 + (16*a + 7)*47 + (10*a + 18)*47^2 + (43*a + 36)*47^3 + (16*a + 14)*47^4+O(47^5) $r_{ 3 }$ $=$ $$2 a + 23 + \left(30 a + 43\right)\cdot 47 + \left(36 a + 21\right)\cdot 47^{2} + \left(3 a + 18\right)\cdot 47^{3} + \left(30 a + 5\right)\cdot 47^{4} +O(47^{5})$$ 2*a + 23 + (30*a + 43)*47 + (36*a + 21)*47^2 + (3*a + 18)*47^3 + (30*a + 5)*47^4+O(47^5) $r_{ 4 }$ $=$ $$24 a + 16 + \left(8 a + 18\right)\cdot 47 + \left(44 a + 34\right)\cdot 47^{2} + \left(39 a + 40\right)\cdot 47^{3} + \left(9 a + 37\right)\cdot 47^{4} +O(47^{5})$$ 24*a + 16 + (8*a + 18)*47 + (44*a + 34)*47^2 + (39*a + 40)*47^3 + (9*a + 37)*47^4+O(47^5) $r_{ 5 }$ $=$ $$23 a + 17 + \left(38 a + 11\right)\cdot 47 + \left(2 a + 20\right)\cdot 47^{2} + \left(7 a + 29\right)\cdot 47^{3} + \left(37 a + 17\right)\cdot 47^{4} +O(47^{5})$$ 23*a + 17 + (38*a + 11)*47 + (2*a + 20)*47^2 + (7*a + 29)*47^3 + (37*a + 17)*47^4+O(47^5)

## 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.