# Properties

 Label 3.13e2_31e2.12t33.2c1 Dimension 3 Group $A_5$ Conductor $13^{2} \cdot 31^{2}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $162409= 13^{2} \cdot 31^{2}$ Artin number field: Splitting field of $f= x^{5} - 31 x^{2} + 31 x + 31$ 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 $\Q_{ 467 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $57 + 167\cdot 467 + 58\cdot 467^{2} + 389\cdot 467^{3} + 407\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 2 }$ $=$ $272 + 204\cdot 467 + 155\cdot 467^{2} + 14\cdot 467^{3} + 376\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 3 }$ $=$ $308 + 20\cdot 467 + 320\cdot 467^{2} + 154\cdot 467^{3} + 164\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 4 }$ $=$ $318 + 174\cdot 467 + 217\cdot 467^{2} + 306\cdot 467^{3} + 317\cdot 467^{4} +O\left(467^{ 5 }\right)$ $r_{ 5 }$ $=$ $446 + 366\cdot 467 + 182\cdot 467^{2} + 69\cdot 467^{3} + 135\cdot 467^{4} +O\left(467^{ 5 }\right)$

### 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.