# Properties

 Label 5.3157729.10t13.a Dimension $5$ Group $S_5$ Conductor $3157729$ Indicator $1$

# Learn more

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $$3157729$$$$\medspace = 1777^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.1777.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Projective image: $S_5$ Projective field: 5.1.1777.1

## 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:
 $r_{ 1 }$ $=$ $$3 a + 8 + \left(11 a + 8\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(13 a + 3\right)\cdot 23^{3} + \left(17 a + 13\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 2 }$ $=$ $$20 a + 14 + \left(11 a + 4\right)\cdot 23 + \left(5 a + 17\right)\cdot 23^{2} + \left(9 a + 13\right)\cdot 23^{3} + \left(5 a + 11\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 3 }$ $=$ $$2 a + 3 + \left(16 a + 17\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(8 a + 3\right)\cdot 23^{3} + \left(18 a + 7\right)\cdot 23^{4} +O(23^{5})$$ $r_{ 4 }$ $=$ $$15 + 14\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O(23^{5})$$ $r_{ 5 }$ $=$ $$21 a + 7 + \left(6 a + 1\right)\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(14 a + 7\right)\cdot 23^{3} + \left(4 a + 12\right)\cdot 23^{4} +O(23^{5})$$

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