Properties

 Label 1.5.4t1.a.a Dimension $1$ Group $C_4$ Conductor $5$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$5$$ Artin number field: Galois closure of $$\Q(\zeta_{5})$$ Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{5}(3,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

Defining polynomial

 $f(x)$ $=$ $x^{4} - x^{3} + x^{2} - x + 1$.

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $2 + 10\cdot 11 + 4\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 8\cdot 11 + 5\cdot 11^{2} + 9\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 + 3\cdot 11 + 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 4 }$ $=$ $8 + 10\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)(3,4)$ $-1$ $1$ $4$ $(1,4,2,3)$ $\zeta_{4}$ $1$ $4$ $(1,3,2,4)$ $-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.