Properties

Label 16.536...144.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $5.368\times 10^{24}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(536\!\cdots\!144\)\(\medspace = 2^{10} \cdot 3^{8} \cdot 19^{14}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.4633831094976.1
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 9.3.4633831094976.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} - x^{7} + 20x^{6} - 30x^{5} - 19x^{4} + 99x^{3} - 91x^{2} + 16x + 3 \) 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:
$r_{ 1 }$ $=$ \( 2 a^{3} + 2 a^{2} + 20 a + 6 + \left(6 a^{3} + 22 a^{2} + 2 a + 20\right)\cdot 23 + \left(18 a^{3} + a^{2} + 3 a + 10\right)\cdot 23^{2} + \left(16 a^{3} + 17 a^{2} + 16 a + 11\right)\cdot 23^{3} + \left(19 a^{3} + 21 a^{2} + 19 a + 6\right)\cdot 23^{4} + \left(6 a^{3} + 15 a^{2} + 21 a + 7\right)\cdot 23^{5} + \left(15 a^{3} + 19 a^{2} + 19 a + 18\right)\cdot 23^{6} + \left(3 a^{3} + 16 a^{2} + 15\right)\cdot 23^{7} + \left(17 a^{3} + 3 a^{2} + 12 a + 18\right)\cdot 23^{8} + \left(17 a^{3} + 21 a^{2} + 2 a + 17\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a^{3} + a^{2} + 14 a + \left(12 a^{3} + 6 a^{2} + 19 a + 15\right)\cdot 23 + \left(21 a^{3} + 3 a^{2} + 16 a + 13\right)\cdot 23^{2} + \left(20 a^{3} + 13 a^{2} + 11 a + 18\right)\cdot 23^{3} + \left(19 a^{3} + 21 a^{2} + 18 a + 8\right)\cdot 23^{4} + \left(15 a^{3} + 4 a^{2} + 4 a + 21\right)\cdot 23^{5} + \left(22 a^{3} + 6 a^{2} + 11 a + 22\right)\cdot 23^{6} + \left(3 a^{3} + 5 a^{2} + 17 a + 2\right)\cdot 23^{7} + \left(21 a^{3} + 22 a^{2} + 18 a\right)\cdot 23^{8} + \left(2 a^{3} + 5 a^{2} + 16 a + 8\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 a^{3} + 19 a^{2} + 9 + \left(22 a^{3} + 10 a + 17\right)\cdot 23 + \left(11 a^{3} + 12 a^{2} + 7 a + 18\right)\cdot 23^{2} + \left(17 a^{3} + 10 a^{2} + 21 a + 21\right)\cdot 23^{3} + \left(18 a^{3} + 2 a^{2} + 19 a + 11\right)\cdot 23^{4} + \left(a^{3} + 12 a^{2} + 19 a + 17\right)\cdot 23^{5} + \left(3 a^{3} + a^{2} + 14 a + 19\right)\cdot 23^{6} + \left(9 a^{3} + 8 a^{2} + 9 a + 5\right)\cdot 23^{7} + \left(19 a^{3} + 6 a^{2} + 8 a + 9\right)\cdot 23^{8} + \left(2 a^{3} + 5 a^{2} + 10 a + 2\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 a^{3} + 18 a + 5 + \left(3 a^{3} + 8 a^{2} + 10 a + 20\right)\cdot 23 + \left(12 a^{3} + 12 a^{2} + 6 a + 9\right)\cdot 23^{2} + \left(9 a^{3} + 15 a + 19\right)\cdot 23^{3} + \left(21 a^{3} + 17 a^{2} + 21 a + 19\right)\cdot 23^{4} + \left(22 a^{3} + 3 a^{2} + a + 6\right)\cdot 23^{5} + \left(11 a^{3} + 10 a^{2} + 2 a + 10\right)\cdot 23^{6} + \left(2 a^{3} + 3 a^{2} + 20 a + 2\right)\cdot 23^{7} + \left(7 a^{3} + 9 a^{2} + 6 a + 5\right)\cdot 23^{8} + \left(18 a^{3} + 2 a^{2} + 13 a + 17\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 18 a^{3} + 19 a^{2} + 6 a + 18 + \left(13 a^{3} + 3 a^{2} + 8 a + 4\right)\cdot 23 + \left(2 a^{3} + 13 a^{2} + 5 a + 17\right)\cdot 23^{2} + \left(14 a^{3} + 9 a + 11\right)\cdot 23^{3} + \left(13 a^{3} + 9 a^{2} + 12 a + 9\right)\cdot 23^{4} + \left(16 a^{3} + 3 a^{2} + 8 a + 6\right)\cdot 23^{5} + \left(16 a^{3} + 8 a^{2} + 2 a + 21\right)\cdot 23^{6} + \left(19 a^{3} + 6 a^{2} + 6 a + 21\right)\cdot 23^{7} + \left(19 a^{3} + 6 a^{2} + 14 a + 14\right)\cdot 23^{8} + \left(13 a^{3} + 10 a^{2} + 3 a + 3\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 14 a^{3} + 13 a^{2} + 18 a + 6 + \left(11 a^{3} + 8 a^{2} + 19 a + 3\right)\cdot 23 + \left(20 a^{3} + a^{2} + a + 9\right)\cdot 23^{2} + \left(4 a^{2} + 13 a + 22\right)\cdot 23^{3} + \left(17 a^{3} + 14 a^{2} + 8 a + 4\right)\cdot 23^{4} + \left(a^{3} + 10 a^{2} + 19 a + 14\right)\cdot 23^{5} + \left(9 a^{3} + 9 a^{2} + 21 a + 13\right)\cdot 23^{6} + \left(20 a^{3} + 5 a^{2} + 11 a + 18\right)\cdot 23^{7} + \left(20 a^{3} + a^{2} + 7 a + 22\right)\cdot 23^{8} + \left(6 a^{3} + 6 a^{2} + 11 a + 3\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 11 a^{3} + a^{2} + 6 a + 12 + \left(13 a^{3} + 14 a^{2} + 15 a + 21\right)\cdot 23 + \left(3 a^{3} + 4 a^{2} + 20 a + 6\right)\cdot 23^{2} + \left(17 a^{3} + 15 a^{2} + 8 a + 2\right)\cdot 23^{3} + \left(15 a^{3} + 16 a^{2} + 18 a + 11\right)\cdot 23^{4} + \left(6 a^{3} + 21 a^{2} + 10 a + 13\right)\cdot 23^{5} + \left(14 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 23^{6} + \left(18 a^{3} + 17 a^{2} + 21 a + 11\right)\cdot 23^{7} + \left(10 a^{3} + 13 a^{2} + 6\right)\cdot 23^{8} + \left(11 a^{3} + 8 a^{2} + 7\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 12 a^{3} + 14 a^{2} + 10 a + 2 + \left(8 a^{3} + 5 a^{2} + 5 a + 18\right)\cdot 23 + \left(a^{3} + 20 a^{2} + 7 a + 17\right)\cdot 23^{2} + \left(18 a^{3} + 7 a^{2} + 19 a + 19\right)\cdot 23^{3} + \left(11 a^{3} + 12 a^{2} + 18 a + 7\right)\cdot 23^{4} + \left(19 a^{3} + 19 a^{2} + 4 a + 16\right)\cdot 23^{5} + \left(21 a^{3} + a^{2} + 7 a + 11\right)\cdot 23^{6} + \left(13 a^{3} + 6 a^{2} + 4 a + 2\right)\cdot 23^{7} + \left(21 a^{3} + 6 a^{2} + 6\right)\cdot 23^{8} + \left(17 a^{3} + 9 a^{2} + 11 a + 22\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 14 + 17\cdot 23 + 10\cdot 23^{2} + 10\cdot 23^{3} + 11\cdot 23^{4} + 11\cdot 23^{5} + 10\cdot 23^{6} + 10\cdot 23^{7} + 8\cdot 23^{8} + 9\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.