# Properties

 Label 1.1015.4t1.a Dimension 1 Group $C_4$ Conductor $5 \cdot 7 \cdot 29$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $1015= 5 \cdot 7 \cdot 29$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 254 x^{2} + 254 x + 12751$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Even 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:
 $r_{ 1 }$ $=$ $38\cdot 41 + 35\cdot 41^{2} + 16\cdot 41^{3} + 19\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $1 + 21\cdot 41 + 19\cdot 41^{2} + 9\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 + 25\cdot 41 + 24\cdot 41^{2} + 34\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $34 + 38\cdot 41 + 41^{2} + 21\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 5 }\right)$

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