Properties

Label 1.5e2.5t1.1c4
Dimension 1
Group $C_5$
Conductor $ 5^{2}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_5$
Conductor:$25= 5^{2} $
Artin number field: Splitting field of $f= x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_5$
Parity: Even
Corresponding Dirichlet character: \(\chi_{25}(16,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 3\cdot 43 + 24\cdot 43^{2} + 5\cdot 43^{3} + 7\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 13\cdot 43 + 19\cdot 43^{2} + 5\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 14\cdot 43 + 26\cdot 43^{2} + 32\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 + 12\cdot 43 + 2\cdot 43^{2} + 38\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 33 + 42\cdot 43 + 13\cdot 43^{2} + 4\cdot 43^{3} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

Character values on conjugacy classes

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