Properties

Label 16.155...769.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $1.557\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(155\!\cdots\!769\)\(\medspace = 3^{14} \cdot 71^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.280155320935227.2
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 9.3.280155320935227.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 2x^{8} + 7x^{7} - 11x^{6} - 8x^{5} - 14x^{4} - 8x^{3} - 104x^{2} + 13x + 87 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: \( x^{4} + 2x^{2} + 66x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 19 + 71\cdot 79 + 56\cdot 79^{2} + 33\cdot 79^{3} + 54\cdot 79^{4} + 75\cdot 79^{5} + 3\cdot 79^{6} + 28\cdot 79^{7} + 25\cdot 79^{8} + 36\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a^{3} + 22 a^{2} + 16 a + 38 + \left(43 a^{3} + a^{2} + 78 a + 28\right)\cdot 79 + \left(15 a^{3} + 19 a^{2} + 9 a + 39\right)\cdot 79^{2} + \left(47 a^{3} + 21 a^{2} + 12 a + 75\right)\cdot 79^{3} + \left(48 a^{3} + 10 a^{2} + 59 a + 1\right)\cdot 79^{4} + \left(73 a^{3} + 26 a^{2} + 10 a + 12\right)\cdot 79^{5} + \left(70 a^{3} + 76 a^{2} + a + 3\right)\cdot 79^{6} + \left(41 a^{3} + 22 a^{2} + 17 a + 40\right)\cdot 79^{7} + \left(52 a^{3} + 51 a^{2} + 28 a + 16\right)\cdot 79^{8} + \left(11 a^{3} + 62 a^{2} + 25 a + 6\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 46 a^{3} + 75 a^{2} + 23 a + 51 + \left(3 a^{3} + 50 a^{2} + 62 a + 22\right)\cdot 79 + \left(12 a^{3} + 24 a^{2} + 47 a + 52\right)\cdot 79^{2} + \left(2 a^{3} + 12 a^{2} + 73 a + 2\right)\cdot 79^{3} + \left(35 a^{3} + 23 a^{2} + 72 a + 69\right)\cdot 79^{4} + \left(9 a^{3} + 52 a^{2} + 21 a + 50\right)\cdot 79^{5} + \left(53 a^{3} + 64 a^{2} + 48 a + 16\right)\cdot 79^{6} + \left(5 a^{3} + 50 a^{2} + 48 a + 31\right)\cdot 79^{7} + \left(42 a^{3} + 29 a^{2} + 31 a + 40\right)\cdot 79^{8} + \left(37 a^{3} + 43 a^{2} + 77 a + 37\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 21 a^{3} + 46 a^{2} + 59 a + 5 + \left(12 a^{3} + 60 a^{2} + 28 a + 26\right)\cdot 79 + \left(69 a^{3} + 46 a^{2} + 71 a + 74\right)\cdot 79^{2} + \left(60 a^{3} + 70 a^{2} + 10 a + 11\right)\cdot 79^{3} + \left(50 a^{3} + 73 a^{2} + 72 a + 15\right)\cdot 79^{4} + \left(78 a^{3} + 41 a^{2} + 74 a\right)\cdot 79^{5} + \left(53 a^{3} + 45 a^{2} + 44 a + 3\right)\cdot 79^{6} + \left(17 a^{3} + 62 a^{2} + 37 a + 66\right)\cdot 79^{7} + \left(48 a^{3} + 69 a^{2} + 50 a + 58\right)\cdot 79^{8} + \left(2 a^{3} + 42 a + 9\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 47 a^{3} + 44 a^{2} + 58 a + 30 + \left(73 a^{3} + 34 a^{2} + 8 a + 35\right)\cdot 79 + \left(58 a^{3} + 19 a^{2} + 23 a + 37\right)\cdot 79^{2} + \left(35 a^{3} + 24 a^{2} + 29 a + 18\right)\cdot 79^{3} + \left(27 a^{3} + 38 a^{2} + 43 a + 25\right)\cdot 79^{4} + \left(53 a^{3} + 48 a^{2} + 19 a + 8\right)\cdot 79^{5} + \left(25 a^{3} + 37 a^{2} + 59 a + 53\right)\cdot 79^{6} + \left(49 a^{3} + 3 a^{2} + 47 a + 11\right)\cdot 79^{7} + \left(23 a^{2} + 63 a + 36\right)\cdot 79^{8} + \left(51 a^{3} + 75 a^{2} + 42 a + 25\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 52 a^{3} + 59 a^{2} + 27 a + 12 + \left(28 a^{3} + 30 a^{2} + 39 a + 57\right)\cdot 79 + \left(60 a^{3} + 32 a^{2} + 44 a + 19\right)\cdot 79^{2} + \left(65 a^{3} + 20 a^{2} + 36 a + 6\right)\cdot 79^{3} + \left(30 a^{3} + 19 a^{2} + 28 a + 40\right)\cdot 79^{4} + \left(9 a^{3} + 7 a^{2} + 51 a + 13\right)\cdot 79^{5} + \left(59 a^{3} + 34 a^{2} + 32 a + 8\right)\cdot 79^{6} + \left(27 a^{3} + 12 a^{2} + 35 a + 40\right)\cdot 79^{7} + \left(24 a^{3} + 58 a^{2} + 48 a + 50\right)\cdot 79^{8} + \left(21 a^{3} + 4 a^{2} + 66 a + 30\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 21 a^{3} + 5 a^{2} + 38 a + 47 + \left(44 a^{3} + 43 a^{2} + 14 a + 53\right)\cdot 79 + \left(28 a^{3} + 57 a^{2} + 77 a + 33\right)\cdot 79^{2} + \left(18 a^{3} + 23 a^{2} + 51 a + 26\right)\cdot 79^{3} + \left(64 a^{3} + 15 a^{2} + a + 45\right)\cdot 79^{4} + \left(75 a^{3} + 18 a^{2} + 15 a + 23\right)\cdot 79^{5} + \left(29 a^{3} + 13 a^{2} + 24 a + 43\right)\cdot 79^{6} + \left(47 a^{3} + 55 a^{2} + 7 a + 6\right)\cdot 79^{7} + \left(47 a^{3} + 28 a^{2} + 67 a + 76\right)\cdot 79^{8} + \left(5 a^{3} + 57 a^{2} + 32 a + 50\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 3 a^{3} + 31 a^{2} + 56 a + 47 + \left(74 a^{3} + 65 a^{2} + 11 a + 7\right)\cdot 79 + \left(12 a^{3} + 59 a^{2} + 32 a + 30\right)\cdot 79^{2} + \left(63 a^{3} + 45 a^{2} + 19 a + 21\right)\cdot 79^{3} + \left(27 a^{3} + 54 a^{2} + 77 a + 4\right)\cdot 79^{4} + \left(75 a^{3} + 3 a^{2} + 20 a + 36\right)\cdot 79^{5} + \left(52 a^{3} + 2 a^{2} + 26\right)\cdot 79^{6} + \left(70 a^{3} + 60 a^{2} + 68 a\right)\cdot 79^{7} + \left(32 a^{3} + 57 a^{2} + 30 a + 78\right)\cdot 79^{8} + \left(43 a^{3} + 10 a^{2} + 23 a + 24\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 44 a^{3} + 34 a^{2} + 39 a + 69 + \left(36 a^{3} + 29 a^{2} + 72 a + 13\right)\cdot 79 + \left(58 a^{3} + 56 a^{2} + 9 a + 51\right)\cdot 79^{2} + \left(22 a^{3} + 18 a^{2} + 3 a + 40\right)\cdot 79^{3} + \left(31 a^{3} + 2 a^{2} + 40 a + 60\right)\cdot 79^{4} + \left(19 a^{3} + 39 a^{2} + 22 a + 16\right)\cdot 79^{5} + \left(49 a^{3} + 42 a^{2} + 26 a\right)\cdot 79^{6} + \left(55 a^{3} + 48 a^{2} + 54 a + 13\right)\cdot 79^{7} + \left(67 a^{3} + 76 a^{2} + 74 a + 13\right)\cdot 79^{8} + \left(63 a^{3} + 60 a^{2} + 4 a + 15\right)\cdot 79^{9} +O(79^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.