Properties

Label 16.232...704.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $2.321\times 10^{25}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(232\!\cdots\!704\)\(\medspace = 2^{34} \cdot 3^{14} \cdot 7^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.16862305517568.2
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 9.3.16862305517568.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 6x^{7} - 6x^{6} - 24x^{5} + 84x^{4} - 150x^{3} + 150x^{2} - 93x + 27 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{4} + 3x^{2} + 40x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.