# Properties

 Label 1.1013.4t1.1c2 Dimension 1 Group $C_4$ Conductor $1013$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $1013$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 127 x^{2} + 2849 x + 26983$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{1013}(45,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 4\cdot 19 + 6\cdot 19^{2} + 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 15\cdot 19 + 14\cdot 19^{2} + 18\cdot 19^{3} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 + 10\cdot 19^{2} + 19^{3} + 12\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $18 + 17\cdot 19 + 6\cdot 19^{2} + 17\cdot 19^{3} + 4\cdot 19^{4} +O\left(19^{ 5 }\right)$

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