Properties

 Label 4.5501.5t5.a.a Dimension 4 Group $S_5$ Conductor $5501$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $5501$ Artin number field: Splitting field of 5.1.5501.1 defined by $f= x^{5} - 2 x^{4} + 2 x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.5501.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.1.5501.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots:
 $r_{ 1 }$ $=$ $25 + 19\cdot 47 + 46\cdot 47^{2} + 37\cdot 47^{3} + 23\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 a + 14 + \left(41 a + 8\right)\cdot 47 + \left(38 a + 2\right)\cdot 47^{2} + \left(44 a + 41\right)\cdot 47^{3} + \left(36 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 a + 16 + \left(45 a + 27\right)\cdot 47 + \left(40 a + 8\right)\cdot 47^{2} + \left(a + 27\right)\cdot 47^{3} + 17 a\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $27 a + 9 + \left(a + 4\right)\cdot 47 + \left(6 a + 45\right)\cdot 47^{2} + \left(45 a + 36\right)\cdot 47^{3} + \left(29 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $38 a + 32 + \left(5 a + 34\right)\cdot 47 + \left(8 a + 38\right)\cdot 47^{2} + \left(2 a + 44\right)\cdot 47^{3} + \left(10 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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.