Properties

Label 1.183.10t1.a.d
Dimension $1$
Group $C_{10}$
Conductor $183$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{10}$
Conductor: \(183\)\(\medspace = 3 \cdot 61 \)
Artin field: 10.0.46584877058339283.1
Galois orbit size: $4$
Smallest permutation container: $C_{10}$
Parity: odd
Dirichlet character: \(\chi_{183}(131,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{10} - x^{9} + 25 x^{8} - 10 x^{7} + 552 x^{6} - 313 x^{5} + 1286 x^{4} + 1321 x^{3} + 1460 x^{2} + 533 x + 169\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \(x^{5} + 5 x + 17\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a^{4} + 13 a^{3} + 6 a^{2} + 7 + \left(17 a^{4} + a^{3} + 18 a^{2} + 5 a + 16\right)\cdot 19 + \left(14 a^{4} + 11 a^{3} + 14 a^{2} + a + 4\right)\cdot 19^{2} + \left(8 a^{4} + 17 a^{3} + 16 a^{2} + 4 a + 12\right)\cdot 19^{3} + \left(6 a^{4} + 10 a^{3} + 11 a^{2} + a + 6\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 a^{4} + 4 a^{3} + 10 a^{2} + 10 a + 14 + \left(14 a^{4} + 12 a^{3} + 10 a + 10\right)\cdot 19 + \left(13 a^{4} + 4 a^{3} + 10 a^{2} + 2 a + 18\right)\cdot 19^{2} + \left(18 a^{4} + 9 a^{3} + 6 a^{2} + 18 a + 17\right)\cdot 19^{3} + \left(14 a^{4} + 6 a^{3} + 9 a^{2} + 14 a + 6\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 8 a^{4} + 11 a^{3} + 10 a^{2} + 12 a + 4 + \left(13 a^{4} + 3 a^{3} + 11 a^{2} + 14 a + 1\right)\cdot 19 + \left(8 a^{4} + 2 a^{3} + 5 a^{2} + 6 a + 18\right)\cdot 19^{2} + \left(8 a^{4} + 8 a^{3} + 13 a + 10\right)\cdot 19^{3} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 7 a + 13\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{4} + 15 a^{3} + 14 a^{2} + 16 a + 7 + \left(2 a^{3} + 14 a^{2} + 10 a + 12\right)\cdot 19 + \left(2 a^{4} + 8 a^{3} + 5 a + 9\right)\cdot 19^{2} + \left(12 a^{4} + 6 a^{3} + 10 a^{2} + 13 a + 10\right)\cdot 19^{3} + \left(14 a^{4} + 13 a^{2} + a + 5\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 9 a^{4} + 15 a^{3} + 4 a^{2} + 11 + \left(9 a^{4} + 16 a^{3} + 9 a^{2} + 16 a + 10\right)\cdot 19 + \left(10 a^{4} + 3 a^{3} + 3 a^{2} + 7 a + 5\right)\cdot 19^{2} + \left(6 a^{4} + 16 a^{3} + 14 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(12 a^{3} + 8 a^{2} + 2 a + 5\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 12 a^{4} + 13 a^{3} + 2 a^{2} + 5 a + 1 + \left(9 a^{4} + 18 a^{3} + 14 a^{2} + 15 a + 5\right)\cdot 19 + \left(16 a^{4} + 15 a^{3} + 16 a^{2} + 3 a + 11\right)\cdot 19^{2} + \left(2 a^{4} + 15 a^{3} + 13 a^{2} + 7\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 6 a^{2} + 17 a + 8\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 16 a^{4} + 14 a^{3} + 5 a^{2} + 6 a + 17 + \left(15 a^{4} + 15 a^{2} + 9 a + 10\right)\cdot 19 + \left(14 a^{4} + 10 a^{3} + 6 a^{2} + 8 a + 4\right)\cdot 19^{2} + \left(17 a^{4} + 16 a^{3} + 4 a^{2} + 11 a + 10\right)\cdot 19^{3} + \left(14 a^{4} + 10 a^{3} + 14 a^{2} + 13 a + 2\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 17 a^{4} + 3 a^{3} + 16 a^{2} + 4 a + 5 + \left(8 a^{4} + 6 a^{3} + 10 a^{2} + 3 a + 8\right)\cdot 19 + \left(16 a^{4} + 8 a^{3} + 7 a^{2} + 13 a + 10\right)\cdot 19^{2} + \left(a^{4} + 5 a^{3} + 11 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(16 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 11\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 17 a^{4} + 6 a^{3} + 15 a^{2} + 15 a + 2 + \left(13 a^{3} + 16 a^{2} + 12 a + 8\right)\cdot 19 + \left(2 a^{4} + 17 a^{3} + 12 a^{2} + 17 a + 10\right)\cdot 19^{2} + \left(17 a^{3} + 2 a^{2} + 8 a + 15\right)\cdot 19^{3} + \left(11 a^{4} + 13 a^{2} + 17 a + 5\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 10 }$ $=$ \( 18 a^{4} + a^{3} + 13 a^{2} + 8 a + 9 + \left(4 a^{4} + 2 a^{2} + 16 a + 11\right)\cdot 19 + \left(14 a^{4} + 13 a^{3} + 16 a^{2} + 8 a + 1\right)\cdot 19^{2} + \left(17 a^{4} + 14 a^{2} + 12 a + 14\right)\cdot 19^{3} + \left(10 a^{4} + 9 a^{3} + 16 a^{2} + 9 a + 9\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

Cycle notation
$(1,5)(2,9)(3,10)(4,6)(7,8)$
$(1,2,3,8,6,5,9,10,7,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 10 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,5)(2,9)(3,10)(4,6)(7,8)$$-1$
$1$$5$$(1,3,6,9,7)(2,8,5,10,4)$$\zeta_{5}$
$1$$5$$(1,6,7,3,9)(2,5,4,8,10)$$\zeta_{5}^{2}$
$1$$5$$(1,9,3,7,6)(2,10,8,4,5)$$\zeta_{5}^{3}$
$1$$5$$(1,7,9,6,3)(2,4,10,5,8)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$
$1$$10$$(1,2,3,8,6,5,9,10,7,4)$$-\zeta_{5}^{3}$
$1$$10$$(1,8,9,4,3,5,7,2,6,10)$$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$
$1$$10$$(1,10,6,2,7,5,3,4,9,8)$$-\zeta_{5}$
$1$$10$$(1,4,7,10,9,5,6,8,3,2)$$-\zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.