# Properties

 Label 1.2e3_13.4t1.2c1 Dimension 1 Group $C_4$ Conductor $2^{3} \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $104= 2^{3} \cdot 13$ Artin number field: Splitting field of $f= x^{4} + 26 x^{2} + 52$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Odd Corresponding Dirichlet character: $$\chi_{104}(5,\cdot)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $2 + 24\cdot 43 + 9\cdot 43^{2} + 3\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 + 21\cdot 43 + 13\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $23 + 21\cdot 43 + 42\cdot 43^{2} + 29\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $41 + 18\cdot 43 + 33\cdot 43^{2} + 39\cdot 43^{3} + 41\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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