Properties

Label 16.512...721.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $5.124\times 10^{24}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(512\!\cdots\!721\)\(\medspace = 3^{26} \cdot 17^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.38483081420787.2
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 9.3.38483081420787.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 3x^{7} - 12x^{6} + 39x^{5} - 45x^{4} + 63x^{3} - 216x^{2} + 153x + 42 \) 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:
$r_{ 1 }$ $=$ \( 41 + 48\cdot 73 + 10\cdot 73^{2} + 61\cdot 73^{3} + 68\cdot 73^{4} + 10\cdot 73^{5} + 44\cdot 73^{6} + 61\cdot 73^{7} + 12\cdot 73^{8} + 45\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a^{3} + 23 a^{2} + 69 + \left(40 a^{3} + 7 a^{2} + 37 a + 50\right)\cdot 73 + \left(18 a^{3} + 39 a^{2} + 24 a + 33\right)\cdot 73^{2} + \left(68 a^{3} + 50 a^{2} + 44 a + 54\right)\cdot 73^{3} + \left(21 a^{3} + 34 a^{2} + 52 a + 14\right)\cdot 73^{4} + \left(47 a^{3} + 10 a^{2} + 21 a + 49\right)\cdot 73^{5} + \left(18 a^{3} + 63 a^{2} + 45 a + 61\right)\cdot 73^{6} + \left(63 a^{3} + 24 a^{2} + 37 a + 61\right)\cdot 73^{7} + \left(68 a^{3} + 56 a^{2} + 18 a + 64\right)\cdot 73^{8} + \left(30 a^{3} + a^{2} + 6 a + 52\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a^{3} + 4 a^{2} + 34 a + 72 + \left(33 a^{3} + 11 a^{2} + 72 a + 51\right)\cdot 73 + \left(8 a^{3} + 26 a^{2} + 59 a + 13\right)\cdot 73^{2} + \left(69 a^{3} + 34 a^{2} + 23 a + 46\right)\cdot 73^{3} + \left(61 a^{3} + 27 a^{2} + 41 a + 64\right)\cdot 73^{4} + \left(59 a^{3} + 48 a^{2} + 34 a + 9\right)\cdot 73^{5} + \left(59 a^{3} + 65 a^{2} + 28 a + 1\right)\cdot 73^{6} + \left(68 a^{3} + 28 a^{2} + 68 a + 60\right)\cdot 73^{7} + \left(39 a^{3} + 37 a^{2} + 68 a + 68\right)\cdot 73^{8} + \left(8 a^{3} + 68 a^{2} + 9 a + 57\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 51 a^{3} + 66 a^{2} + 42 a + 71 + \left(17 a^{3} + 61 a^{2} + 14 a + 57\right)\cdot 73 + \left(28 a^{3} + 5 a^{2} + 22 a + 28\right)\cdot 73^{2} + \left(44 a^{3} + 67 a^{2} + 64 a + 60\right)\cdot 73^{3} + \left(32 a^{3} + 25 a^{2} + 3 a + 27\right)\cdot 73^{4} + \left(58 a^{3} + 5 a^{2} + 66 a + 38\right)\cdot 73^{5} + \left(27 a^{3} + 6 a^{2} + 18 a + 62\right)\cdot 73^{6} + \left(39 a^{3} + 37 a^{2} + 37 a + 32\right)\cdot 73^{7} + \left(58 a^{3} + 34 a^{2} + 50 a + 40\right)\cdot 73^{8} + \left(49 a^{3} + 71 a^{2} + 18 a + 15\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 7 a^{3} + 30 a^{2} + 55 a + 21 + \left(58 a^{3} + 9 a^{2} + 49 a + 70\right)\cdot 73 + \left(62 a^{3} + 41 a^{2} + 18 a + 6\right)\cdot 73^{2} + \left(35 a^{3} + 41 a^{2} + 29 a + 21\right)\cdot 73^{3} + \left(58 a^{3} + 44 a^{2} + 23 a + 56\right)\cdot 73^{4} + \left(33 a^{3} + 34 a^{2} + 24 a + 46\right)\cdot 73^{5} + \left(62 a^{3} + 7 a^{2} + 32 a + 11\right)\cdot 73^{6} + \left(51 a^{3} + 34 a^{2} + 14 a + 46\right)\cdot 73^{7} + \left(63 a^{3} + 23 a^{2} + 13 a + 6\right)\cdot 73^{8} + \left(29 a^{3} + 56 a^{2} + 4 a + 53\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 60 a^{3} + 18 a^{2} + 59 a + 65 + \left(51 a^{2} + 14 a + 66\right)\cdot 73 + \left(20 a^{3} + 65 a^{2} + 36 a + 15\right)\cdot 73^{2} + \left(18 a^{3} + 64 a^{2} + 25 a + 41\right)\cdot 73^{3} + \left(52 a^{3} + 31 a^{2} + 52 a + 24\right)\cdot 73^{4} + \left(34 a^{3} + 34 a^{2} + 6 a + 4\right)\cdot 73^{5} + \left(62 a^{3} + 37 a^{2} + 9 a + 18\right)\cdot 73^{6} + \left(47 a^{3} + 11 a^{2} + 47 a + 38\right)\cdot 73^{7} + \left(63 a^{3} + 57 a^{2} + 25 a + 71\right)\cdot 73^{8} + \left(18 a^{3} + 46 a^{2} + 66 a + 51\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 8 a^{3} + 44 a^{2} + 60 a + 29 + \left(67 a^{3} + 44 a^{2} + 19 a\right)\cdot 73 + \left(5 a^{3} + 56 a^{2} + 59 a + 5\right)\cdot 73^{2} + \left(11 a^{3} + 36 a^{2} + a + 37\right)\cdot 73^{3} + \left(49 a^{3} + 65 a^{2} + 33 a + 50\right)\cdot 73^{4} + \left(48 a^{3} + 38 a^{2} + 55 a + 48\right)\cdot 73^{5} + \left(26 a^{3} + 50 a^{2} + 9 a + 20\right)\cdot 73^{6} + \left(59 a^{3} + 21 a^{2} + 10 a + 43\right)\cdot 73^{7} + \left(3 a^{3} + 15 a^{2} + 36 a + 53\right)\cdot 73^{8} + \left(35 a^{3} + 8 a^{2} + 6 a + 31\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 28 a^{3} + 39 a^{2} + 45 a + 57 + \left(14 a^{3} + 25 a^{2} + 6 a + 65\right)\cdot 73 + \left(6 a^{3} + 35 a^{2} + 63 a + 68\right)\cdot 73^{2} + \left(15 a^{3} + 36 a^{2} + 11 a + 44\right)\cdot 73^{3} + \left(39 a^{3} + 53 a^{2} + 37 a + 14\right)\cdot 73^{4} + \left(5 a^{3} + 22 a^{2} + 51 a + 72\right)\cdot 73^{5} + \left(37 a^{3} + 39 a^{2} + 72 a + 60\right)\cdot 73^{6} + \left(68 a^{3} + 72 a^{2} + 23 a + 6\right)\cdot 73^{7} + \left(27 a^{3} + 70 a^{2} + 51 a + 69\right)\cdot 73^{8} + \left(46 a^{3} + 25 a^{2} + 54 a + 18\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 57 a^{3} + 68 a^{2} + 70 a + 16 + \left(60 a^{3} + 7 a^{2} + 3 a + 25\right)\cdot 73 + \left(68 a^{3} + 22 a^{2} + 8 a + 35\right)\cdot 73^{2} + \left(29 a^{3} + 33 a^{2} + 18 a + 71\right)\cdot 73^{3} + \left(49 a^{3} + 8 a^{2} + 48 a + 42\right)\cdot 73^{4} + \left(3 a^{3} + 24 a^{2} + 31 a + 11\right)\cdot 73^{5} + \left(70 a^{3} + 22 a^{2} + 2 a + 11\right)\cdot 73^{6} + \left(38 a^{3} + 61 a^{2} + 53 a + 14\right)\cdot 73^{7} + \left(38 a^{3} + 69 a^{2} + 27 a + 50\right)\cdot 73^{8} + \left(72 a^{3} + 12 a^{2} + 52 a + 37\right)\cdot 73^{9} +O(73^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.