# Properties

 Label 1.2e2_17.4t1.1 Dimension 1 Group $C_4$ Conductor $2^{2} \cdot 17$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $68= 2^{2} \cdot 17$ Artin number field: Splitting field of $f= x^{4} + 17 x^{2} + 68$ 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 5.
Roots:
 $r_{ 1 }$ $=$ $5 + 10\cdot 13 + 6\cdot 13^{2} + 8\cdot 13^{3} + 10\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 2\cdot 13 + 5\cdot 13^{2} + 4\cdot 13^{3} + 8\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 + 10\cdot 13 + 7\cdot 13^{2} + 8\cdot 13^{3} + 4\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $8 + 2\cdot 13 + 6\cdot 13^{2} + 4\cdot 13^{3} + 2\cdot 13^{4} +O\left(13^{ 5 }\right)$

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