# Properties

 Label 1.37.3t1.a.a Dimension $1$ Group $C_3$ Conductor $37$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$37$$ Artin field: 3.3.1369.1 Galois orbit size: $2$ Smallest permutation container: $C_3$ Parity: even Dirichlet character: $$\chi_{37}(10,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{3} - x^{2} - 12 x - 11$$  .

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 value $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $\zeta_{3}$ $1$ $3$ $(1,3,2)$ $-\zeta_{3} - 1$

The blue line marks the conjugacy class containing complex conjugation.