# Properties

 Label 1.101.5t1.a.b Dimension 1 Group $C_5$ Conductor $101$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $101$ Artin number field: Splitting field of 5.5.104060401.1 defined by $f= x^{5} - x^{4} - 40 x^{3} - 93 x^{2} - 21 x + 17$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_5$ Parity: Even Corresponding Dirichlet character: $$\chi_{101}(95,\cdot)$$ 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:
 $r_{ 1 }$ $=$ $6 + 30\cdot 41 + 13\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $21 + 36\cdot 41 + 33\cdot 41^{2} + 22\cdot 41^{3} + 19\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $24 + 35\cdot 41 + 15\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $34 + 8\cdot 41 + 26\cdot 41^{2} + 29\cdot 41^{3} + 10\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 5 }$ $=$ $39 + 11\cdot 41 + 7\cdot 41^{2} + 12\cdot 41^{3} + 4\cdot 41^{4} +O\left(41^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

 Cycle notation $(1,5,2,4,3)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $1$ $1$ $5$ $(1,5,2,4,3)$ $\zeta_{5}^{2}$ $1$ $5$ $(1,2,3,5,4)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,4,5,3,2)$ $\zeta_{5}$ $1$ $5$ $(1,3,4,2,5)$ $\zeta_{5}^{3}$
The blue line marks the conjugacy class containing complex conjugation.