# Properties

 Label 4.2665.5t5.b Dimension $4$ Group $S_5$ Conductor $2665$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $$2665$$$$\medspace = 5 \cdot 13 \cdot 41$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.2665.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Projective image: $S_5$ Projective field: 5.1.2665.1

## Galois action

### Roots of defining polynomial

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:
 $r_{ 1 }$ $=$ $$2 + 21\cdot 23 + 4\cdot 23^{2} + 23^{3} + 5\cdot 23^{4} +O(23^{5})$$ $r_{ 2 }$ $=$ $$10 + 12\cdot 23 + 15\cdot 23^{2} + 5\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})$$ $r_{ 3 }$ $=$ $$9 + 9\cdot 23 + 2\cdot 23^{2} + 11\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})$$ $r_{ 4 }$ $=$ $$13 a + \left(3 a + 16\right)\cdot 23 + \left(17 a + 7\right)\cdot 23^{2} + \left(17 a + 16\right)\cdot 23^{3} + \left(9 a + 4\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 5 }$ $=$ $$10 a + 3 + \left(19 a + 10\right)\cdot 23 + \left(5 a + 15\right)\cdot 23^{2} + \left(5 a + 11\right)\cdot 23^{3} + \left(13 a + 6\right)\cdot 23^{4} +O(23^{5})$$

### 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 values $c1$ $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.