# Properties

 Label 1.3_337.4t1.1 Dimension 1 Group $C_4$ Conductor $3 \cdot 337$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $1011= 3 \cdot 337$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 211 x^{2} + 1032 x + 17628$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $7\cdot 13 + 5\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} + 8\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 2 }$ $=$ $4 + 5\cdot 13 + 11\cdot 13^{2} + 8\cdot 13^{3} + 11\cdot 13^{4} + 10\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 3 }$ $=$ $11 + 9\cdot 13 + 2\cdot 13^{2} + 3\cdot 13^{3} + 2\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 4 }$ $=$ $12 + 3\cdot 13 + 6\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\right)$

### 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 values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,3)(2,4)$ $-1$ $-1$ $1$ $4$ $(1,2,3,4)$ $\zeta_{4}$ $-\zeta_{4}$ $1$ $4$ $(1,4,3,2)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.