Properties

Label 16.303...929.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $3.036\times 10^{24}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(303\!\cdots\!929\)\(\medspace = 3^{18} \cdot 97^{8}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.17964142659.1
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 9.3.17964142659.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a^{3} + 20 a^{2} + 6 a + 6 + \left(17 a^{3} + 2 a^{2} + 14 a + 8\right)\cdot 23 + \left(3 a^{3} + 18 a^{2} + 19 a\right)\cdot 23^{2} + \left(18 a^{3} + a^{2} + a + 6\right)\cdot 23^{3} + \left(13 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 23^{4} + \left(11 a^{3} + 15 a^{2} + 12 a + 20\right)\cdot 23^{5} + \left(a^{3} + 14 a^{2} + 7 a + 3\right)\cdot 23^{6} + \left(7 a^{3} + 22 a^{2} + 16 a + 17\right)\cdot 23^{7} + \left(21 a^{3} + 21 a^{2} + 18 a + 21\right)\cdot 23^{8} + \left(21 a^{3} + 12 a^{2} + 16 a + 2\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 a^{3} + 12 a^{2} + 22 a + 7 + \left(21 a^{3} + 16 a^{2} + 11 a + 4\right)\cdot 23 + \left(14 a^{3} + 6 a^{2} + 9 a + 21\right)\cdot 23^{2} + \left(20 a^{3} + 9 a^{2} + 18 a + 1\right)\cdot 23^{3} + \left(5 a^{3} + 17 a^{2} + 12 a + 20\right)\cdot 23^{4} + \left(7 a^{3} + 18 a^{2} + 15 a + 12\right)\cdot 23^{5} + \left(18 a^{3} + 5 a^{2} + 15 a + 5\right)\cdot 23^{6} + \left(22 a^{3} + 7 a^{2} + 11 a + 4\right)\cdot 23^{7} + \left(22 a^{3} + 14 a^{2} + 4 a + 4\right)\cdot 23^{8} + \left(22 a^{3} + a^{2} + 18 a + 9\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 a^{3} + 19 a^{2} + 9 a + 8 + \left(a^{3} + 21 a^{2} + a + 13\right)\cdot 23 + \left(16 a^{3} + 21 a^{2} + 22 a + 8\right)\cdot 23^{2} + \left(2 a^{3} + 11 a^{2} + 7 a + 20\right)\cdot 23^{3} + \left(9 a^{3} + 11 a^{2} + 5 a + 16\right)\cdot 23^{4} + \left(a^{3} + 11 a^{2} + 16 a + 4\right)\cdot 23^{5} + \left(12 a^{3} + 3 a^{2} + 7 a + 22\right)\cdot 23^{6} + \left(18 a^{3} + 19 a^{2} + 7 a + 12\right)\cdot 23^{7} + \left(2 a^{3} + 18 a^{2} + 6 a + 16\right)\cdot 23^{8} + \left(18 a^{3} + 19 a^{2} + 3 a + 18\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 a^{3} + 3 a^{2} + 16 a + 12 + \left(15 a^{3} + 19 a^{2} + 5 a + 10\right)\cdot 23 + \left(15 a^{3} + 7 a^{2} + 7 a + 22\right)\cdot 23^{2} + \left(14 a^{3} + 4 a^{2} + 17 a + 11\right)\cdot 23^{3} + \left(10 a^{3} + 2 a^{2} + 3 a\right)\cdot 23^{4} + \left(6 a^{3} + 12 a^{2} + 15 a + 17\right)\cdot 23^{5} + \left(13 a^{3} + 6 a^{2} + 22 a + 15\right)\cdot 23^{6} + \left(16 a^{3} + 10 a^{2} + a + 7\right)\cdot 23^{7} + \left(10 a^{3} + 2 a^{2} + 3 a + 8\right)\cdot 23^{8} + \left(18 a^{3} + 3 a^{2} + 12 a + 19\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 19 a^{3} + 11 a^{2} + 2 a + 13 + \left(2 a^{3} + 17 a^{2} + 6 a + 16\right)\cdot 23 + \left(19 a^{3} + 14 a^{2} + 21 a + 6\right)\cdot 23^{2} + \left(6 a^{3} + 19 a^{2} + 9 a + 10\right)\cdot 23^{3} + \left(22 a^{3} + a^{2} + 22 a\right)\cdot 23^{4} + \left(a^{3} + 9 a^{2} + 14 a + 15\right)\cdot 23^{5} + \left(4 a^{3} + 9 a^{2} + 17 a + 3\right)\cdot 23^{6} + \left(14 a^{3} + 16 a^{2} + 16 a + 10\right)\cdot 23^{7} + \left(18 a^{3} + 21 a^{2} + 9 a + 10\right)\cdot 23^{8} + \left(10 a^{3} + 12 a^{2} + 3 a + 13\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a^{3} + 4 a^{2} + 18 a + 17 + \left(20 a^{3} + 2 a^{2} + 7 a + 18\right)\cdot 23 + \left(22 a^{3} + 22 a^{2} + 22 a + 2\right)\cdot 23^{2} + \left(14 a^{3} + 16 a^{2} + 6 a + 18\right)\cdot 23^{3} + \left(18 a^{3} + 12 a^{2} + 21 a + 9\right)\cdot 23^{4} + \left(21 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 23^{5} + \left(8 a^{3} + 11 a^{2} + 4 a + 1\right)\cdot 23^{6} + \left(4 a^{3} + 16 a^{2} + 10 a + 10\right)\cdot 23^{7} + \left(4 a^{3} + 6 a^{2} + a + 19\right)\cdot 23^{8} + \left(11 a^{3} + 6 a^{2} + 16 a\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 2 a^{3} + 19 a^{2} + 6 a + 11 + \left(16 a^{3} + 21 a^{2} + 18 a + 1\right)\cdot 23 + \left(3 a^{3} + 20 a^{2} + 19 a + 21\right)\cdot 23^{2} + \left(21 a^{3} + 22 a^{2} + 19 a + 22\right)\cdot 23^{3} + \left(2 a^{3} + 3 a^{2} + 5 a + 13\right)\cdot 23^{4} + \left(6 a^{3} + 11 a^{2} + 15 a + 16\right)\cdot 23^{5} + \left(22 a^{3} + 13 a^{2} + 11 a + 4\right)\cdot 23^{6} + \left(17 a^{3} + 19 a^{2} + 17 a + 1\right)\cdot 23^{7} + \left(9 a^{3} + 14 a^{2} + 22 a + 2\right)\cdot 23^{8} + \left(17 a^{3} + 18\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 19 + 22\cdot 23 + 16\cdot 23^{3} + 11\cdot 23^{4} + 19\cdot 23^{5} + 18\cdot 23^{6} + 23^{7} + 17\cdot 23^{8} +O(23^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 4 a^{2} + 13 a + 2 + \left(20 a^{3} + 13 a^{2} + 3 a + 19\right)\cdot 23 + \left(18 a^{3} + 2 a^{2} + 16 a + 7\right)\cdot 23^{2} + \left(15 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 23^{3} + \left(8 a^{3} + 15 a^{2} + 5 a + 16\right)\cdot 23^{4} + \left(12 a^{3} + 6 a^{2} + 22 a + 21\right)\cdot 23^{5} + \left(11 a^{3} + 4 a^{2} + 4 a + 15\right)\cdot 23^{6} + \left(13 a^{3} + 3 a^{2} + 10 a + 3\right)\cdot 23^{7} + \left(a^{3} + 14 a^{2} + 2 a + 15\right)\cdot 23^{8} + \left(17 a^{3} + 11 a^{2} + 21 a + 8\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.