# Properties

 Label 1.111.4t1.a.a Dimension 1 Group $C_4$ Conductor $3 \cdot 37$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $111= 3 \cdot 37$ Artin number field: Splitting field of 4.4.455877.1 defined by $f= x^{4} - x^{3} - 32 x^{2} + 30 x + 123$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Even Corresponding Dirichlet character: $$\chi_{111}(68,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,4,3,2)$ $(1,3)(2,4)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,3)(2,4)$ $-1$ $1$ $4$ $(1,4,3,2)$ $\zeta_{4}$ $1$ $4$ $(1,2,3,4)$ $-\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.