# Properties

 Label 3.202500.12t33.a Dimension 3 Group $A_5$ Conductor $2^{2} \cdot 3^{4} \cdot 5^{4}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $202500= 2^{2} \cdot 3^{4} \cdot 5^{4}$ Artin number field: Splitting field of $f= x^{5} - 15 x^{3} - 65 x^{2} - 90 x - 57$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.126562500.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 487 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $128 + 250\cdot 487 + 389\cdot 487^{2} + 431\cdot 487^{3} + 390\cdot 487^{4} +O\left(487^{ 5 }\right)$ $r_{ 2 }$ $=$ $186 + 407\cdot 487 + 388\cdot 487^{2} + 261\cdot 487^{3} + 146\cdot 487^{4} +O\left(487^{ 5 }\right)$ $r_{ 3 }$ $=$ $291 + 264\cdot 487 + 269\cdot 487^{2} + 279\cdot 487^{3} + 412\cdot 487^{4} +O\left(487^{ 5 }\right)$ $r_{ 4 }$ $=$ $421 + 213\cdot 487 + 229\cdot 487^{2} + 153\cdot 487^{3} + 76\cdot 487^{4} +O\left(487^{ 5 }\right)$ $r_{ 5 }$ $=$ $435 + 324\cdot 487 + 183\cdot 487^{2} + 334\cdot 487^{3} + 434\cdot 487^{4} +O\left(487^{ 5 }\right)$

### 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 values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.