# Properties

 Label 1.10069.3t1.a.a Dimension 1 Group $C_3$ Conductor $10069$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $10069$ Artin number field: Splitting field of 3.3.101384761.1 defined by $f= x^{3} - x^{2} - 3356 x + 75331$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{10069}(5363,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $7 + 10\cdot 37 + 5\cdot 37^{2} + 20\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $14 + 37 + 10\cdot 37^{2} + 26\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $17 + 25\cdot 37 + 21\cdot 37^{2} + 27\cdot 37^{3} +O\left(37^{ 5 }\right)$

### 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.