Properties

Label 1.11.5t1.1c1
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: \(\chi_{11}(3,\cdot)\)

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

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