# Properties

 Label 1.5_13.4t1.4c1 Dimension 1 Group $C_4$ Conductor $5 \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $65= 5 \cdot 13$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 16 x^{2} - 16 x + 61$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{65}(38,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $12 + 43\cdot 59 + 3\cdot 59^{2} + 23\cdot 59^{3} + 9\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $14 + 10\cdot 59 + 22\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $43 + 10\cdot 59 + 27\cdot 59^{2} + 59^{3} + 17\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 4 }$ $=$ $50 + 53\cdot 59 + 5\cdot 59^{2} + 12\cdot 59^{3} + 5\cdot 59^{4} +O\left(59^{ 5 }\right)$

### 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.