Properties

 Label 1.11.5t1.1c3 Dimension 1 Group $C_5$ Conductor $11$ Root number not computed Frobenius-Schur indicator 0

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Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $11$ Artin number field: Splitting field of $f= x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ over $\Q$ Size of Galois orbit: 4 Smallest containing permutation representation: $C_5$ Parity: Even Corresponding Dirichlet character: $$\chi_{11}(5,\cdot)$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $9 + 20\cdot 23 + 11\cdot 23^{2} + 6\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 + 11\cdot 23^{2} + 17\cdot 23^{3} + 20\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 + 4\cdot 23 + 7\cdot 23^{2} + 14\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $17 + 3\cdot 23 + 4\cdot 23^{2} + 4\cdot 23^{3} + 21\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $19 + 16\cdot 23 + 11\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O\left(23^{ 5 }\right)$

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 $1$ $1$ $()$ $1$ $1$ $5$ $(1,5,2,4,3)$ $\zeta_{5}^{3}$ $1$ $5$ $(1,2,3,5,4)$ $\zeta_{5}$ $1$ $5$ $(1,4,5,3,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,3,4,2,5)$ $\zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.