Properties

 Label 1.61.5t1.a.b Dimension $1$ Group $C_5$ Conductor $61$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $$61$$ Artin field: 5.5.13845841.1 Galois orbit size: $4$ Smallest permutation container: $C_5$ Parity: even Dirichlet character: $$\chi_{61}(34,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} - 24 x^{3} + 17 x^{2} + 41 x + 13$$  .

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

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

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $1$ $1$ $5$ $(1,4,5,2,3)$ $\zeta_{5}^{2}$ $1$ $5$ $(1,5,3,4,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,2,4,3,5)$ $\zeta_{5}$ $1$ $5$ $(1,3,2,5,4)$ $\zeta_{5}^{3}$

The blue line marks the conjugacy class containing complex conjugation.