# Properties

 Label 1.101.4t1.a.b Dimension 1 Group $C_4$ Conductor $101$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $101$ Artin number field: Splitting field of 4.0.1030301.1 defined by $f= x^{4} - x^{3} + 13 x^{2} - 19 x + 361$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{101}(10,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $11 + 30\cdot 31 + 7\cdot 31^{2} + 15\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $13 + 29\cdot 31 + 12\cdot 31^{2} + 18\cdot 31^{3} + 15\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $15 + 4\cdot 31 + 12\cdot 31^{2} + 9\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $24 + 28\cdot 31 + 28\cdot 31^{2} + 18\cdot 31^{3} + 13\cdot 31^{4} +O\left(31^{ 5 }\right)$

### 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.