# Properties

 Label 4.2297.5t5.a.a Dimension $4$ Group $S_5$ Conductor $2297$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $$2297$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 5.1.2297.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Determinant: 1.2297.2t1.a.a Projective image: $S_5$ Projective stem field: Galois closure of 5.1.2297.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$10 + 7\cdot 17 + 8\cdot 17^{2} + 6\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})$$ 10 + 7*17 + 8*17^2 + 6*17^3 + 12*17^4+O(17^5) $r_{ 2 }$ $=$ $$10 a + \left(8 a + 2\right)\cdot 17 + \left(6 a + 12\right)\cdot 17^{2} + \left(9 a + 6\right)\cdot 17^{3} + \left(11 a + 7\right)\cdot 17^{4} +O(17^{5})$$ 10*a + (8*a + 2)*17 + (6*a + 12)*17^2 + (9*a + 6)*17^3 + (11*a + 7)*17^4+O(17^5) $r_{ 3 }$ $=$ $$7 a + 10 + 8 a\cdot 17 + \left(10 a + 10\right)\cdot 17^{2} + \left(7 a + 9\right)\cdot 17^{3} + \left(5 a + 9\right)\cdot 17^{4} +O(17^{5})$$ 7*a + 10 + 8*a*17 + (10*a + 10)*17^2 + (7*a + 9)*17^3 + (5*a + 9)*17^4+O(17^5) $r_{ 4 }$ $=$ $$6 a + 13 + \left(2 a + 13\right)\cdot 17 + \left(11 a + 5\right)\cdot 17^{2} + \left(5 a + 8\right)\cdot 17^{3} + \left(15 a + 14\right)\cdot 17^{4} +O(17^{5})$$ 6*a + 13 + (2*a + 13)*17 + (11*a + 5)*17^2 + (5*a + 8)*17^3 + (15*a + 14)*17^4+O(17^5) $r_{ 5 }$ $=$ $$11 a + 2 + \left(14 a + 10\right)\cdot 17 + \left(5 a + 14\right)\cdot 17^{2} + \left(11 a + 2\right)\cdot 17^{3} + \left(a + 7\right)\cdot 17^{4} +O(17^{5})$$ 11*a + 2 + (14*a + 10)*17 + (5*a + 14)*17^2 + (11*a + 2)*17^3 + (a + 7)*17^4+O(17^5)

## 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.