# Properties

 Label 6.18927429625.20t30.a.a Dimension $6$ Group $\PGL(2,5)$ Conductor $18927429625$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $\PGL(2,5)$ Conductor: $$18927429625$$$$\medspace = 5^{3} \cdot 13^{3} \cdot 41^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 6.2.18927429625.2 Galois orbit size: $1$ Smallest permutation container: 20T30 Parity: even Determinant: 1.2665.2t1.a.a Projective image: $S_5$ Projective stem field: 6.2.18927429625.2

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 17 x^{4} + 117 x^{3} - 188 x^{2} + 23 x + 594$$  .

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $$x^{2} + 49 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$16 + 36\cdot 53 + 29\cdot 53^{2} + 11\cdot 53^{3} + 11\cdot 53^{4} +O(53^{5})$$ $r_{ 2 }$ $=$ $$19 a + 23 + \left(40 a + 13\right)\cdot 53 + \left(14 a + 30\right)\cdot 53^{2} + \left(47 a + 43\right)\cdot 53^{3} + \left(24 a + 51\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 3 }$ $=$ $$34 a + 46 + \left(12 a + 49\right)\cdot 53 + \left(38 a + 48\right)\cdot 53^{2} + \left(5 a + 5\right)\cdot 53^{3} + \left(28 a + 51\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 4 }$ $=$ $$39 a + \left(32 a + 18\right)\cdot 53 + \left(17 a + 40\right)\cdot 53^{2} + \left(15 a + 6\right)\cdot 53^{3} + \left(49 a + 48\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 5 }$ $=$ $$14 a + 50 + \left(20 a + 3\right)\cdot 53 + \left(35 a + 25\right)\cdot 53^{2} + \left(37 a + 50\right)\cdot 53^{3} + \left(3 a + 17\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 6 }$ $=$ $$25 + 37\cdot 53 + 37\cdot 53^{2} + 40\cdot 53^{3} + 31\cdot 53^{4} +O(53^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.