Properties

Label 16.652...281.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $6.523\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(652\!\cdots\!281\)\(\medspace = 3^{18} \cdot 17^{14}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.475099770627.2
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 9.3.475099770627.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} - 3x^{5} + 6x^{3} + 33x^{2} + 27x - 7 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{4} + 16x^{2} + 56x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 32 + 49\cdot 73 + 51\cdot 73^{2} + 22\cdot 73^{3} + 22\cdot 73^{4} + 19\cdot 73^{5} + 27\cdot 73^{6} + 41\cdot 73^{7} + 31\cdot 73^{8} + 30\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a^{3} + 2 a^{2} + 45 a + 72 + \left(51 a^{3} + 13 a^{2} + 32 a + 23\right)\cdot 73 + \left(47 a^{3} + 13 a^{2} + 36 a + 66\right)\cdot 73^{2} + \left(2 a^{3} + 16 a^{2} + 25 a + 63\right)\cdot 73^{3} + \left(25 a^{3} + 6 a^{2} + 7 a + 68\right)\cdot 73^{4} + \left(30 a^{3} + 24 a^{2} + 60 a + 36\right)\cdot 73^{5} + \left(40 a^{3} + 15 a^{2} + 11 a + 37\right)\cdot 73^{6} + \left(14 a^{3} + 51 a^{2} + 30 a + 10\right)\cdot 73^{7} + \left(33 a^{3} + 26 a^{2} + 64 a + 67\right)\cdot 73^{8} + \left(58 a^{3} + 67 a^{2} + 4 a + 19\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a^{3} + 10 a^{2} + 46 a + 46 + \left(66 a^{3} + 66 a^{2} + 43\right)\cdot 73 + \left(21 a^{3} + 13 a^{2} + 50 a + 10\right)\cdot 73^{2} + \left(14 a^{3} + 67 a^{2} + 25 a + 12\right)\cdot 73^{3} + \left(53 a^{3} + 6 a^{2} + 12 a + 16\right)\cdot 73^{4} + \left(56 a^{3} + 39 a + 4\right)\cdot 73^{5} + \left(14 a^{3} + 32 a^{2} + 68 a + 43\right)\cdot 73^{6} + \left(20 a^{3} + 24 a^{2} + 41 a + 33\right)\cdot 73^{7} + \left(7 a^{3} + 5 a^{2} + 51 a + 48\right)\cdot 73^{8} + \left(10 a^{3} + 55 a^{2} + 47 a + 7\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 a^{3} + 25 a^{2} + 11 a + 72 + \left(61 a^{3} + 37 a^{2} + 40 a + 18\right)\cdot 73 + \left(33 a^{3} + 65 a^{2} + 68 a + 51\right)\cdot 73^{2} + \left(62 a^{3} + 5 a^{2} + 19 a + 49\right)\cdot 73^{3} + \left(25 a^{3} + 32 a^{2} + 41 a + 51\right)\cdot 73^{4} + \left(17 a^{3} + 18 a^{2} + 58 a + 38\right)\cdot 73^{5} + \left(62 a^{3} + 65 a^{2} + 26 a + 2\right)\cdot 73^{6} + \left(8 a^{3} + 71 a + 65\right)\cdot 73^{7} + \left(38 a^{3} + 71 a^{2} + 65 a + 33\right)\cdot 73^{8} + \left(63 a^{3} + 35 a^{2} + 21 a + 27\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 23 a^{3} + 42 a^{2} + 25 a + 2 + \left(12 a^{3} + 48 a^{2} + 55 a + 27\right)\cdot 73 + \left(27 a^{3} + 26 a^{2} + 34 a + 44\right)\cdot 73^{2} + \left(18 a^{3} + 47 a^{2} + 28 a + 62\right)\cdot 73^{3} + \left(23 a^{3} + 70 a^{2} + 39 a + 31\right)\cdot 73^{4} + \left(53 a^{3} + 43 a^{2} + 35 a + 1\right)\cdot 73^{5} + \left(10 a^{3} + 20 a^{2} + 7 a + 37\right)\cdot 73^{6} + \left(58 a^{3} + 70 a^{2} + 18 a + 62\right)\cdot 73^{7} + \left(67 a^{3} + 42 a^{2} + 42 a + 33\right)\cdot 73^{8} + \left(64 a^{3} + 41 a^{2} + 50 a + 58\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 17 a^{3} + 22 a^{2} + 2 a + 11 + \left(55 a^{3} + 36 a^{2} + 48 a + 9\right)\cdot 73 + \left(a^{3} + 27 a^{2} + 68 a + 3\right)\cdot 73^{2} + \left(40 a^{3} + 4 a^{2} + 10 a + 6\right)\cdot 73^{3} + \left(39 a^{3} + 70 a^{2} + 58 a + 21\right)\cdot 73^{4} + \left(21 a^{3} + 52 a^{2} + 67 a + 44\right)\cdot 73^{5} + \left(58 a^{3} + 71 a^{2} + 3 a + 71\right)\cdot 73^{6} + \left(48 a^{3} + 64 a^{2} + 5 a + 25\right)\cdot 73^{7} + \left(3 a^{3} + 11 a^{2} + 48 a + 22\right)\cdot 73^{8} + \left(41 a^{3} + 9 a^{2} + 13 a + 64\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 31 a^{3} + 39 a^{2} + 53 a + 5 + \left(46 a^{3} + 30 a^{2} + 64 a + 31\right)\cdot 73 + \left(a^{3} + 18 a^{2} + 63 a + 71\right)\cdot 73^{2} + \left(16 a^{3} + 58 a^{2} + 10 a + 12\right)\cdot 73^{3} + \left(28 a^{3} + 62 a^{2} + 68 a + 71\right)\cdot 73^{4} + \left(37 a^{3} + 68 a^{2} + 51 a + 33\right)\cdot 73^{5} + \left(32 a^{3} + 26 a^{2} + 61 a + 17\right)\cdot 73^{6} + \left(62 a^{3} + 5 a^{2} + 68 a + 49\right)\cdot 73^{7} + \left(28 a^{3} + 29 a^{2} + 54 a + 49\right)\cdot 73^{8} + \left(36 a^{3} + 14 a^{2} + 6 a + 56\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 68 a^{3} + 23 a^{2} + 10 a + 61 + \left(8 a^{3} + 4 a^{2} + 55 a + 41\right)\cdot 73 + \left(52 a^{3} + 63 a^{2} + 23 a + 69\right)\cdot 73^{2} + \left(48 a^{3} + 51 a^{2} + 30 a + 58\right)\cdot 73^{3} + \left(29 a^{3} + 3 a^{2} + 22 a + 57\right)\cdot 73^{4} + \left(41 a^{3} + 66 a^{2} + 40 a + 42\right)\cdot 73^{5} + \left(39 a^{3} + 41 a^{2} + 68 a + 31\right)\cdot 73^{6} + \left(49 a^{3} + 4 a^{2} + 53 a + 51\right)\cdot 73^{7} + \left(22 a^{3} + 26 a^{2} + 48 a + 47\right)\cdot 73^{8} + \left(26 a^{3} + 56 a^{2} + 58 a + 14\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 41 a^{3} + 56 a^{2} + 27 a + 67 + \left(63 a^{3} + 55 a^{2} + 68 a + 46\right)\cdot 73 + \left(32 a^{3} + 63 a^{2} + 18 a + 69\right)\cdot 73^{2} + \left(16 a^{3} + 40 a^{2} + 67 a + 2\right)\cdot 73^{3} + \left(67 a^{3} + 39 a^{2} + 42 a + 24\right)\cdot 73^{4} + \left(33 a^{3} + 17 a^{2} + 11 a + 70\right)\cdot 73^{5} + \left(33 a^{3} + 18 a^{2} + 43 a + 23\right)\cdot 73^{6} + \left(29 a^{3} + 70 a^{2} + 2 a + 25\right)\cdot 73^{7} + \left(17 a^{3} + 5 a^{2} + 62 a + 30\right)\cdot 73^{8} + \left(64 a^{3} + 12 a^{2} + 14 a + 12\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.