Properties

Label 1.101.5t1.1
Dimension 1
Group $C_5$
Conductor $ 101 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_5$
Conductor:$101 $
Artin number field: Splitting field of $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

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

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$ $c2$ $c3$ $c4$
$1$ $1$ $()$ $1$ $1$ $1$ $1$
$1$ $5$ $(1,5,2,4,3)$ $\zeta_{5}$ $\zeta_{5}^{2}$ $\zeta_{5}^{3}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$
$1$ $5$ $(1,2,3,5,4)$ $\zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}$ $\zeta_{5}^{3}$
$1$ $5$ $(1,4,5,3,2)$ $\zeta_{5}^{3}$ $\zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{2}$
$1$ $5$ $(1,3,4,2,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{3}$ $\zeta_{5}^{2}$ $\zeta_{5}$
The blue line marks the conjugacy class containing complex conjugation.