# Properties

 Label 1.29.4t1.1c2 Dimension 1 Group $C_4$ Conductor $29$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $29$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 4 x^{2} - 20 x + 23$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{29}(12,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $15\cdot 23 + 11\cdot 23^{2} + 16\cdot 23^{3} + 5\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 10\cdot 23 + 21\cdot 23^{2} + 4\cdot 23^{3} + 19\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $8 + 11\cdot 23^{2} + 9\cdot 23^{3} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $10 + 20\cdot 23 + 23^{2} + 15\cdot 23^{3} + 20\cdot 23^{4} +O\left(23^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,2)(3,4)$ $(1,3,2,4)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)(3,4)$ $-1$ $1$ $4$ $(1,3,2,4)$ $-\zeta_{4}$ $1$ $4$ $(1,4,2,3)$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.