# Properties

 Label 1.2e4.4t1.2c2 Dimension 1 Group $C_4$ Conductor $2^{4}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $16= 2^{4}$ Artin number field: Splitting field of $f= x^{4} + 4 x^{2} + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{16}(3,\cdot)$$

## Galois action

### Roots of defining polynomial

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

### 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.