# Properties

 Label 1.10003.3t1.b.b Dimension 1 Group $C_3$ Conductor $7 \cdot 1429$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $10003= 7 \cdot 1429$ Artin number field: Splitting field of 3.3.100060009.2 defined by $f= x^{3} - x^{2} - 3334 x + 39271$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{10003}(5051,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $5 + 7\cdot 23 + 11\cdot 23^{2} + 9\cdot 23^{3} + 6\cdot 23^{4} + 14\cdot 23^{5} +O\left(23^{ 6 }\right)$ $r_{ 2 }$ $=$ $9 + 10\cdot 23 + 12\cdot 23^{2} + 21\cdot 23^{3} + 8\cdot 23^{4} + 17\cdot 23^{5} +O\left(23^{ 6 }\right)$ $r_{ 3 }$ $=$ $10 + 5\cdot 23 + 22\cdot 23^{2} + 14\cdot 23^{3} + 7\cdot 23^{4} + 14\cdot 23^{5} +O\left(23^{ 6 }\right)$

### 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 $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,3)$ $-\zeta_{3} - 1$ $1$ $3$ $(1,3,2)$ $\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.