# Properties

 Label 1.5_31.4t1.1c2 Dimension 1 Group $C_4$ Conductor $5 \cdot 31$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $155= 5 \cdot 31$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 39 x^{2} + 39 x + 281$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Even Corresponding Dirichlet character: $$\chi_{155}(92,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 10 + 4\cdot 29 + 21\cdot 29^{2} + 9\cdot 29^{3} + 22\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 2 } &= 13 + 21\cdot 29 + 28\cdot 29^{2} + 20\cdot 29^{3} + 19\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 3 } &= 14 + 3\cdot 29 + 3\cdot 29^{2} + 20\cdot 29^{3} + 20\cdot 29^{4} +O\left(29^{ 5 }\right) \\ r_{ 4 } &= 22 + 28\cdot 29 + 4\cdot 29^{2} + 7\cdot 29^{3} + 24\cdot 29^{4} +O\left(29^{ 5 }\right) \\ \end{aligned}

### 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 $1$ $1$ $()$ $1$ $1$ $2$ $(1,3)(2,4)$ $-1$ $1$ $4$ $(1,2,3,4)$ $-\zeta_{4}$ $1$ $4$ $(1,4,3,2)$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.