Properties

 Label 5.3e3_19e3_2617e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $3^{3} \cdot 19^{3} \cdot 2617^{3}$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $3319217678593809= 3^{3} \cdot 19^{3} \cdot 2617^{3}$ Artin number field: Splitting field of $f= x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.3_19_2617.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $x^{2} + 7 x + 2$
Roots: \begin{aligned} r_{ 1 } &= 10 a + 8 + \left(10 a + 3\right)\cdot 11 + \left(3 a + 9\right)\cdot 11^{2} + \left(6 a + 2\right)\cdot 11^{3} +O\left(11^{ 5 }\right) \\ r_{ 2 } &= 3 a + 2 + 7\cdot 11 + \left(7 a + 7\right)\cdot 11^{2} + \left(a + 9\right)\cdot 11^{3} + 6\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 3 } &= 5 + 11 + 10\cdot 11^{2} + 8\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 4 } &= 8 a + 3 + \left(10 a + 5\right)\cdot 11 + \left(3 a + 2\right)\cdot 11^{2} + \left(9 a + 9\right)\cdot 11^{3} + \left(10 a + 5\right)\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 5 } &= a + 4 + 4\cdot 11 + \left(7 a + 3\right)\cdot 11^{2} + \left(4 a + 2\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O\left(11^{ 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 $1$ $1$ $()$ $5$ $10$ $2$ $(1,2)$ $-1$ $15$ $2$ $(1,2)(3,4)$ $1$ $20$ $3$ $(1,2,3)$ $-1$ $30$ $4$ $(1,2,3,4)$ $1$ $24$ $5$ $(1,2,3,4,5)$ $0$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.