Properties

Label 16.182...281.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $1.827\times 10^{26}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(182\!\cdots\!281\)\(\medspace = 3^{28} \cdot 41^{8}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.988941019347.2
Galois orbit size: $1$
Smallest permutation container: 24T1334
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $\AGL(2,3)$
Projective stem field: Galois closure of 9.3.988941019347.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 3x^{7} - 6x^{6} + 12x^{5} - 21x^{4} + 39x^{3} - 39x^{2} + 3x + 10 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{4} + 7x^{2} + 10x + 3 \) Copy content Toggle raw display

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

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$16$
$9$$2$$(1,8)(3,9)(4,6)(5,7)$$0$
$36$$2$$(1,2)(3,5)(4,7)$$0$
$8$$3$$(1,3,4)(2,5,7)(6,9,8)$$-2$
$24$$3$$(1,6,7)(4,5,8)$$-2$
$48$$3$$(1,5,6)(2,8,4)(3,7,9)$$1$
$54$$4$$(1,9,8,3)(4,5,6,7)$$0$
$72$$6$$(1,2,6,9,7,3)(4,8)$$0$
$72$$6$$(1,5,4,2,3,7)(6,9,8)$$0$
$54$$8$$(1,2,9,8,3,6,7,5)$$0$
$54$$8$$(1,6,9,5,3,2,7,8)$$0$

The blue line marks the conjugacy class containing complex conjugation.