# Properties

 Label 5.12064136250375.6t14.a.a Dimension 5 Group $S_5$ Conductor $3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 139^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $12064136250375= 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 139^{3}$ Artin number field: Splitting field of 5.3.22935.1 defined by $f= x^{5} + x^{3} - 2 x^{2} - 4 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Odd Determinant: 1.22935.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.3.22935.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots:
 $r_{ 1 }$ $=$ $7 a + 4 + \left(11 a + 10\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(2 a + 5\right)\cdot 13^{3} + \left(a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 a + 11 + \left(a + 1\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + \left(11 a + 8\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $3 a + 5 + \left(12 a + 3\right)\cdot 13 + 10 a\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(10 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 + 10\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 5\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 5 }$ $=$ $10 a + 8 + 12\cdot 13 + \left(2 a + 11\right)\cdot 13^{2} + 9 a\cdot 13^{3} + \left(2 a + 11\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

### 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.