Properties

Label 1.200.10t1.a.d
Dimension $1$
Group $C_{10}$
Conductor $200$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{10}$
Conductor: \(200\)\(\medspace = 2^{3} \cdot 5^{2}\)
Artin field: Galois closure of 10.0.5000000000000000.1
Galois orbit size: $4$
Smallest permutation container: $C_{10}$
Parity: odd
Dirichlet character: \(\chi_{200}(171,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{10} - 10x^{8} - 10x^{7} + 80x^{6} + 118x^{5} + 305x^{4} + 560x^{3} + 1410x^{2} + 260x + 2393 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{5} + 10x^{2} + 9 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a^{4} + a^{3} + 10 a^{2} + 2 a + 3 + \left(5 a^{4} + 5 a^{3} + 4 a^{2} + 10\right)\cdot 11 + \left(2 a^{4} + 6 a^{3} + 3 a^{2} + 4 a\right)\cdot 11^{2} + \left(2 a^{3} + a^{2} + 9 a + 3\right)\cdot 11^{3} + \left(5 a^{2} + 3 a + 8\right)\cdot 11^{4} + \left(2 a^{4} + 8 a^{3} + 2 a^{2} + 10 a + 10\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 a^{4} + 6 a^{3} + a^{2} + 8 a + \left(4 a^{4} + a^{3} + 8 a^{2} + 6 a\right)\cdot 11 + \left(7 a^{4} + 9 a^{3} + 2 a^{2} + a + 6\right)\cdot 11^{2} + \left(5 a^{4} + 2 a^{3} + 5 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(10 a^{4} + 4 a^{3} + 9 a^{2} + 2 a + 10\right)\cdot 11^{4} + \left(a^{4} + 9 a^{3} + 6 a^{2} + 6 a + 7\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a^{4} + 8 a^{3} + 10 a + 1 + \left(8 a^{4} + 2 a^{3} + 9 a + 7\right)\cdot 11 + \left(7 a^{4} + 8 a^{3} + 2 a^{2}\right)\cdot 11^{2} + \left(2 a^{4} + 5 a^{3} + 5 a^{2} + 10 a\right)\cdot 11^{3} + \left(9 a^{4} + 5 a^{3} + 6 a^{2} + 9 a + 7\right)\cdot 11^{4} + \left(9 a^{4} + 3 a^{3} + 5\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a^{4} + 8 a^{3} + 10 a + 7 + \left(8 a^{4} + 2 a^{3} + 9 a + 3\right)\cdot 11 + \left(7 a^{4} + 8 a^{3} + 2 a^{2} + 10\right)\cdot 11^{2} + \left(2 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 2\right)\cdot 11^{3} + \left(9 a^{4} + 5 a^{3} + 6 a^{2} + 9 a + 4\right)\cdot 11^{4} + \left(9 a^{4} + 3 a^{3} + 3\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a^{4} + 8 a^{3} + 10 a^{2} + a + 1 + \left(9 a^{4} + 5 a^{3} + 9 a^{2} + 5 a + 3\right)\cdot 11 + \left(9 a^{4} + 8 a^{3} + 4 a^{2} + a + 2\right)\cdot 11^{2} + \left(2 a^{4} + 7 a^{3} + 4 a^{2} + 10 a + 1\right)\cdot 11^{3} + \left(7 a^{4} + 9 a^{3} + 6 a^{2} + 10\right)\cdot 11^{4} + \left(a^{4} + 2 a^{3} + 9 a + 1\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 5 a^{4} + 6 a^{3} + a^{2} + 8 a + 6 + \left(4 a^{4} + a^{3} + 8 a^{2} + 6 a + 7\right)\cdot 11 + \left(7 a^{4} + 9 a^{3} + 2 a^{2} + a + 4\right)\cdot 11^{2} + \left(5 a^{4} + 2 a^{3} + 5 a^{2} + 7 a + 7\right)\cdot 11^{3} + \left(10 a^{4} + 4 a^{3} + 9 a^{2} + 2 a + 7\right)\cdot 11^{4} + \left(a^{4} + 9 a^{3} + 6 a^{2} + 6 a + 5\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( a^{4} + 8 a^{3} + 10 a^{2} + a + 7 + \left(9 a^{4} + 5 a^{3} + 9 a^{2} + 5 a + 10\right)\cdot 11 + \left(9 a^{4} + 8 a^{3} + 4 a^{2} + a\right)\cdot 11^{2} + \left(2 a^{4} + 7 a^{3} + 4 a^{2} + 10 a + 4\right)\cdot 11^{3} + \left(7 a^{4} + 9 a^{3} + 6 a^{2} + 7\right)\cdot 11^{4} + \left(a^{4} + 2 a^{3} + 9 a + 10\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 5 a^{4} + 10 a^{3} + a^{2} + a + 2 + \left(5 a^{4} + 6 a^{3} + 10 a^{2} + 10\right)\cdot 11 + \left(5 a^{4} + 8 a^{2} + 3 a + 9\right)\cdot 11^{2} + \left(10 a^{4} + 3 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 11^{3} + \left(5 a^{4} + 2 a^{3} + 5 a^{2} + 4 a + 9\right)\cdot 11^{4} + \left(6 a^{4} + 9 a^{3} + 6 a + 6\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 5 a^{4} + 10 a^{3} + a^{2} + a + 8 + \left(5 a^{4} + 6 a^{3} + 10 a^{2} + 6\right)\cdot 11 + \left(5 a^{4} + 8 a^{2} + 3 a + 8\right)\cdot 11^{2} + \left(10 a^{4} + 3 a^{3} + 5 a^{2} + 7 a + 8\right)\cdot 11^{3} + \left(5 a^{4} + 2 a^{3} + 5 a^{2} + 4 a + 6\right)\cdot 11^{4} + \left(6 a^{4} + 9 a^{3} + 6 a + 4\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 8 a^{4} + a^{3} + 10 a^{2} + 2 a + 9 + \left(5 a^{4} + 5 a^{3} + 4 a^{2} + 6\right)\cdot 11 + \left(2 a^{4} + 6 a^{3} + 3 a^{2} + 4 a + 10\right)\cdot 11^{2} + \left(2 a^{3} + a^{2} + 9 a + 5\right)\cdot 11^{3} + \left(5 a^{2} + 3 a + 5\right)\cdot 11^{4} + \left(2 a^{4} + 8 a^{3} + 2 a^{2} + 10 a + 8\right)\cdot 11^{5} +O(11^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.