Properties

 Label 4.2869.5t5.a Dimension 4 Group $S_5$ Conductor $19 \cdot 151$ Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $2869= 19 \cdot 151$ Artin number field: Splitting field of $f= x^{5} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Projective image: $S_5$ Projective field: Galois closure of 5.1.2869.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $x^{2} + 63 x + 2$
Roots:
 $r_{ 1 }$ $=$ $37 a + 11 + \left(52 a + 51\right)\cdot 67 + \left(a + 63\right)\cdot 67^{2} + \left(31 a + 51\right)\cdot 67^{3} + 43\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $34 a + 41 + \left(44 a + 28\right)\cdot 67 + \left(12 a + 3\right)\cdot 67^{2} + \left(18 a + 16\right)\cdot 67^{3} + \left(36 a + 60\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 + 59\cdot 67 + 38\cdot 67^{2} + 16\cdot 67^{3} + 29\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 a + 25 + \left(14 a + 23\right)\cdot 67 + \left(65 a + 18\right)\cdot 67^{2} + \left(35 a + 40\right)\cdot 67^{3} + \left(66 a + 14\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 5 }$ $=$ $33 a + 43 + \left(22 a + 38\right)\cdot 67 + \left(54 a + 9\right)\cdot 67^{2} + \left(48 a + 9\right)\cdot 67^{3} + \left(30 a + 53\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

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 values $c1$ $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.