# Properties

 Label 1.10003.3t1.a Dimension $1$ Group $C_3$ Conductor $10003$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $$10003$$$$\medspace = 7 \cdot 1429$$ Artin number field: Galois closure of 3.3.100060009.1 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 $\Q_{ 17 }$ to precision 7.
Roots:
 $r_{ 1 }$ $=$ $7\cdot 17 + 10\cdot 17^{2} + 13\cdot 17^{3} + 13\cdot 17^{4} + 16\cdot 17^{5} + 14\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 2 }$ $=$ $2 + 12\cdot 17 + 15\cdot 17^{2} + 14\cdot 17^{3} + 12\cdot 17^{4} + 11\cdot 17^{5} + 6\cdot 17^{6} +O\left(17^{ 7 }\right)$ $r_{ 3 }$ $=$ $16 + 14\cdot 17 + 7\cdot 17^{2} + 5\cdot 17^{3} + 7\cdot 17^{4} + 5\cdot 17^{5} + 12\cdot 17^{6} +O\left(17^{ 7 }\right)$

### 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 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.