# Properties

 Label 1.119.4t1.a.b Dimension 1 Group $C_4$ Conductor $7 \cdot 17$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $119= 7 \cdot 17$ Artin number field: Splitting field of 4.0.240737.1 defined by $f= x^{4} - x^{3} + 28 x^{2} + x + 239$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{119}(13,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $36 + 28\cdot 59 + 13\cdot 59^{2} + 12\cdot 59^{3} + 39\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $39 + 47\cdot 59 + 43\cdot 59^{2} + 10\cdot 59^{3} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $48 + 59 + 17\cdot 59^{2} + 16\cdot 59^{3} + 10\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 4 }$ $=$ $55 + 39\cdot 59 + 43\cdot 59^{2} + 19\cdot 59^{3} + 9\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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