Properties

 Label 4.3e3_877e3.10t12.1c1 Dimension 4 Group $S_5$ Conductor $3^{3} \cdot 877^{3}$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $18212205591= 3^{3} \cdot 877^{3}$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + x^{2} - 2 x + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Odd Determinant: 1.3_877.2t1.1c1

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: \begin{aligned} r_{ 1 } &= 14 a + 14 + \left(13 a + 46\right)\cdot 47 + \left(9 a + 41\right)\cdot 47^{2} + \left(31 a + 30\right)\cdot 47^{3} + \left(5 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right) \\ r_{ 2 } &= 33 a + 42 + \left(33 a + 11\right)\cdot 47 + 37 a\cdot 47^{2} + \left(15 a + 37\right)\cdot 47^{3} + \left(41 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right) \\ r_{ 3 } &= 39 a + 11 + \left(22 a + 27\right)\cdot 47 + \left(45 a + 17\right)\cdot 47^{2} + \left(17 a + 7\right)\cdot 47^{3} + \left(24 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right) \\ r_{ 4 } &= 8 a + 42 + \left(24 a + 33\right)\cdot 47 + \left(a + 38\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right) \\ r_{ 5 } &= 34 + 21\cdot 47 + 42\cdot 47^{2} + 20\cdot 47^{3} + 33\cdot 47^{4} +O\left(47^{ 5 }\right) \\ \end{aligned}

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.