# Properties

 Label 1.112.4t1.b.b Dimension 1 Group $C_4$ Conductor $2^{4} \cdot 7$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $112= 2^{4} \cdot 7$ Artin number field: Splitting field of 4.0.100352.5 defined by $f= x^{4} + 28 x^{2} + 98$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{112}(13,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $8 + 21\cdot 41 + 31\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 + 20\cdot 41 + 12\cdot 41^{2} + 5\cdot 41^{3} + 11\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $21 + 20\cdot 41 + 28\cdot 41^{2} + 35\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $33 + 19\cdot 41 + 9\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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