# Properties

 Label 3.5e2_71e2.12t33.1c1 Dimension 3 Group $A_5$ Conductor $5^{2} \cdot 71^{2}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $126025= 5^{2} \cdot 71^{2}$ Artin number field: Splitting field of $f= x^{5} - x^{4} + 3 x^{3} - 4 x^{2} + 5 x - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $232 + 411\cdot 457 + 331\cdot 457^{2} + 218\cdot 457^{3} + 88\cdot 457^{4} +O\left(457^{ 5 }\right)$ $r_{ 2 }$ $=$ $265 + 412\cdot 457 + 390\cdot 457^{2} + 376\cdot 457^{3} + 347\cdot 457^{4} +O\left(457^{ 5 }\right)$ $r_{ 3 }$ $=$ $434 + 441\cdot 457 + 117\cdot 457^{2} + 448\cdot 457^{3} + 125\cdot 457^{4} +O\left(457^{ 5 }\right)$ $r_{ 4 }$ $=$ $444 + 338\cdot 457 + 281\cdot 457^{2} + 19\cdot 457^{3} + 138\cdot 457^{4} +O\left(457^{ 5 }\right)$ $r_{ 5 }$ $=$ $454 + 222\cdot 457 + 248\cdot 457^{2} + 307\cdot 457^{3} + 213\cdot 457^{4} +O\left(457^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.