# Properties

 Label 5.23e3_9419e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $23^{3} \cdot 9419^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $10167118812786853= 23^{3} \cdot 9419^{3}$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} + x - 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.23_9419.2t1.1c1

## 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 } &= 4 a + 5 + 3\cdot 11 + \left(10 a + 7\right)\cdot 11^{2} + \left(8 a + 10\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 2 } &= 7 a + 10 + 10 a\cdot 11 + 3\cdot 11^{2} + \left(2 a + 3\right)\cdot 11^{3} + \left(2 a + 1\right)\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 3 } &= a + 5 + 5\cdot 11 + \left(3 a + 8\right)\cdot 11^{2} + \left(6 a + 6\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 4 } &= 6 + 7\cdot 11 + 4\cdot 11^{2} + 5\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right) \\ r_{ 5 } &= 10 a + 9 + \left(10 a + 4\right)\cdot 11 + \left(7 a + 9\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} +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.