Properties

Label 1.33.10t1.b.d
Dimension $1$
Group $C_{10}$
Conductor $33$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_{10}$
Conductor: \(33\)\(\medspace = 3 \cdot 11 \)
Artin field: Galois closure of \(\Q(\zeta_{33})^+\)
Galois orbit size: $4$
Smallest permutation container: $C_{10}$
Parity: even
Dirichlet character: \(\chi_{33}(8,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{10} - x^{9} - 10x^{8} + 10x^{7} + 34x^{6} - 34x^{5} - 43x^{4} + 43x^{3} + 12x^{2} - 12x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{5} + x + 14 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.