# Properties

 Label 1.37.3t1.a Dimension $1$ Group $C_3$ Conductor $37$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$37$$ Artin number field: Galois closure of 3.3.1369.1 Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$10\cdot 11 + 10\cdot 11^{2} + 9\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})$$ $r_{ 2 }$ $=$ $$4 + 2\cdot 11 + 11^{2} + 3\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})$$ $r_{ 3 }$ $=$ $$8 + 9\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})$$

### Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

 Cycle notation $(1,2,3)$

### Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }, r_{ 3 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $3$ $(1,2,3)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.