# Properties

 Label 3.2e2_5e2_19e2.12t33.1c2 Dimension 3 Group $A_5$ Conductor $2^{2} \cdot 5^{2} \cdot 19^{2}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $36100= 2^{2} \cdot 5^{2} \cdot 19^{2}$ Artin number field: Splitting field of $f= x^{5} - x^{4} - 11 x^{3} + 6 x^{2} + 64 x - 74$ 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 $\Q_{ 557 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $14 + 251\cdot 557 + 288\cdot 557^{2} + 375\cdot 557^{3} + 218\cdot 557^{4} +O\left(557^{ 5 }\right)$ $r_{ 2 }$ $=$ $235 + 435\cdot 557 + 247\cdot 557^{2} + 321\cdot 557^{3} + 239\cdot 557^{4} +O\left(557^{ 5 }\right)$ $r_{ 3 }$ $=$ $375 + 224\cdot 557 + 116\cdot 557^{2} + 75\cdot 557^{3} + 92\cdot 557^{4} +O\left(557^{ 5 }\right)$ $r_{ 4 }$ $=$ $506 + 262\cdot 557 + 255\cdot 557^{2} + 489\cdot 557^{3} + 304\cdot 557^{4} +O\left(557^{ 5 }\right)$ $r_{ 5 }$ $=$ $542 + 496\cdot 557 + 205\cdot 557^{2} + 409\cdot 557^{3} + 258\cdot 557^{4} +O\left(557^{ 5 }\right)$

### 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 $1$ $1$ $()$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $12$ $5$ $(1,2,3,4,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $12$ $5$ $(1,3,4,5,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.