# Properties

 Label 1.13_31.3t1.1c1 Dimension 1 Group $C_3$ Conductor $13 \cdot 31$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_3$ Conductor: $403= 13 \cdot 31$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 134 x - 209$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_3$ Parity: Even Corresponding Dirichlet character: $$\chi_{403}(191,\cdot)$$

## Galois action

### Roots of defining polynomial

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

### 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.