# Properties

 Label 4.5611284433.10t12.b.a Dimension $4$ Group $\PGL(2,5)$ Conductor $5611284433$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $\PGL(2,5)$ Conductor: $$5611284433$$$$\medspace = 1777^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 6.2.5611284433.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Determinant: 1.1777.2t1.a.a Projective image: $S_5$ Projective stem field: 6.2.5611284433.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 3 x^{5} + 8 x^{4} + 45 x^{3} - 211 x^{2} + 293 x - 121$$  .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21 x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$11 a + 4 + \left(20 a + 9\right)\cdot 23 + \left(19 a + 19\right)\cdot 23^{2} + 5\cdot 23^{3} + \left(20 a + 10\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 2 }$ $=$ $$17 + 19\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})$$ $r_{ 3 }$ $=$ $$12 a + 3 + \left(2 a + 16\right)\cdot 23 + \left(3 a + 15\right)\cdot 23^{2} + \left(22 a + 10\right)\cdot 23^{3} + \left(2 a + 3\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 4 }$ $=$ $$10 a + 8 + \left(20 a + 19\right)\cdot 23 + \left(19 a + 10\right)\cdot 23^{2} + \left(4 a + 16\right)\cdot 23^{3} + \left(13 a + 16\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 5 }$ $=$ $$13 a + 5 + \left(2 a + 4\right)\cdot 23 + \left(3 a + 7\right)\cdot 23^{2} + \left(18 a + 6\right)\cdot 23^{3} + \left(9 a + 15\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 6 }$ $=$ $$12 + 18\cdot 23^{2} + 11\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})$$

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,5)(2,3)(4,6)$ $(1,6,5,2,3,4)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $10$ $2$ $(1,5)(2,3)(4,6)$ $-2$ $15$ $2$ $(1,2)(3,5)$ $0$ $20$ $3$ $(1,5,3)(2,4,6)$ $1$ $30$ $4$ $(2,6,3,5)$ $0$ $24$ $5$ $(1,2,4,5,6)$ $-1$ $20$ $6$ $(1,6,5,2,3,4)$ $1$

The blue line marks the conjugacy class containing complex conjugation.