Properties

Label 16.152...561.24t2912.a.a
Dimension $16$
Group $S_3\wr S_3$
Conductor $1.523\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $S_3\wr S_3$
Conductor: \(152\!\cdots\!561\)\(\medspace = 3^{16} \cdot 29^{12}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.134573648589.1
Galois orbit size: $1$
Smallest permutation container: 24T2912
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_3^3.S_4.C_2$
Projective stem field: Galois closure of 9.1.134573648589.1

Defining polynomial

$f(x)$$=$ \( x^{9} + 9x^{7} - 5x^{6} + 27x^{5} - 30x^{4} + 31x^{3} - 45x^{2} + 12x + 9 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{3} + 6x + 65 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$16$
$9$$2$$(2,3)$$0$
$18$$2$$(2,5)(3,6)(7,9)$$0$
$27$$2$$(1,4)(2,3)(5,6)$$0$
$27$$2$$(2,3)(5,6)$$0$
$54$$2$$(1,5)(2,3)(4,6)(7,8)$$0$
$6$$3$$(1,4,8)$$-8$
$8$$3$$(1,4,8)(2,3,9)(5,6,7)$$-2$
$12$$3$$(1,4,8)(5,6,7)$$4$
$72$$3$$(1,2,5)(3,6,4)(7,8,9)$$-2$
$54$$4$$(2,6,3,5)(7,9)$$0$
$162$$4$$(1,2,4,3)(6,7)(8,9)$$0$
$36$$6$$(1,4,8)(2,5)(3,6)(7,9)$$0$
$36$$6$$(1,3,4,9,8,2)$$0$
$36$$6$$(1,4,8)(2,3)$$0$
$36$$6$$(1,4,8)(2,3)(5,6,7)$$0$
$54$$6$$(1,8,4)(2,3)(5,6)$$0$
$72$$6$$(1,4,8)(2,5,3,6,9,7)$$0$
$108$$6$$(1,6,4,7,8,5)(2,3)$$0$
$216$$6$$(1,2,6,4,3,5)(7,8,9)$$0$
$144$$9$$(1,3,6,4,9,7,8,2,5)$$1$
$108$$12$$(1,4,8)(2,6,3,5)(7,9)$$0$

The blue line marks the conjugacy class containing complex conjugation.