# Properties

 Label 1.7_11_13.4t1.1c1 Dimension 1 Group $C_4$ Conductor $7 \cdot 11 \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $1001= 7 \cdot 11 \cdot 13$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 249 x^{2} + 251 x + 4943$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{1001}(538,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $25 + 35\cdot 43 + 11\cdot 43^{2} + 29\cdot 43^{3} + 11\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $30 + 5\cdot 43 + 29\cdot 43^{2} + 37\cdot 43^{3} + 12\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $34 + 4\cdot 43 + 40\cdot 43^{2} + 23\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $41 + 39\cdot 43 + 4\cdot 43^{2} + 38\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$

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