Properties

Label 16.392...625.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $3.925\times 10^{24}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(392\!\cdots\!625\)\(\medspace = 3^{18} \cdot 5^{8} \cdot 11^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.21793521976875.2
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 9.3.21793521976875.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 16x^{6} - 33x^{5} - 36x^{4} + 120x^{3} + 99x^{2} - 297x + 77 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{4} + 2x^{2} + 11x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a^{3} + 16 a^{2} + 5 a + 18 + \left(14 a^{3} + 18 a^{2} + 12 a + 11\right)\cdot 19 + \left(5 a^{3} + 8 a^{2} + 8 a\right)\cdot 19^{2} + \left(11 a^{3} + 5 a^{2} + 11 a + 18\right)\cdot 19^{3} + \left(3 a^{3} + 15 a^{2} + 14 a + 8\right)\cdot 19^{4} + \left(13 a^{3} + 5 a^{2} + 2 a + 17\right)\cdot 19^{5} + \left(17 a^{3} + 5 a^{2} + 8 a + 8\right)\cdot 19^{6} + \left(17 a^{3} + 9 a^{2} + 12 a + 17\right)\cdot 19^{7} + \left(11 a^{3} + 11 a^{2} + 16 a + 5\right)\cdot 19^{8} + \left(5 a^{2} + 16 a + 10\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 a^{3} + 6 a^{2} + 8 a + 9 + \left(10 a^{3} + 16 a^{2} + 11 a + 10\right)\cdot 19 + \left(2 a^{3} + 6 a^{2} + 5 a + 16\right)\cdot 19^{2} + \left(16 a^{3} + 16 a + 13\right)\cdot 19^{3} + \left(14 a^{3} + 13 a^{2} + a + 17\right)\cdot 19^{4} + \left(6 a^{3} + 2 a^{2} + a + 3\right)\cdot 19^{5} + \left(16 a^{3} + 13 a^{2} + 18 a\right)\cdot 19^{6} + \left(a^{3} + 13 a^{2} + 14 a + 13\right)\cdot 19^{7} + \left(5 a^{3} + 4 a^{2} + 15 a + 11\right)\cdot 19^{8} + \left(11 a^{3} + 7 a^{2} + 18 a + 7\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 10\cdot 19 + 2\cdot 19^{2} + 3\cdot 19^{3} + 7\cdot 19^{4} + 18\cdot 19^{5} + 19^{6} + 15\cdot 19^{7} + 5\cdot 19^{8} + 11\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 a^{3} + 7 a^{2} + 6 a + 1 + \left(6 a^{3} + 8 a^{2} + 13 a + 8\right)\cdot 19 + \left(6 a^{3} + 11 a^{2} + 11 a + 3\right)\cdot 19^{2} + \left(6 a^{3} + 11 a^{2} + 8 a + 2\right)\cdot 19^{3} + \left(3 a^{3} + 16 a^{2} + 15 a + 8\right)\cdot 19^{4} + \left(17 a^{3} + 12 a^{2} + 16 a\right)\cdot 19^{5} + \left(2 a^{3} + 2 a^{2} + 12 a + 3\right)\cdot 19^{6} + \left(3 a^{3} + 18 a^{2} + 2 a + 4\right)\cdot 19^{7} + \left(2 a^{3} + 2 a^{2} + 2\right)\cdot 19^{8} + \left(2 a^{3} + 14 a + 17\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a^{3} + 4 a^{2} + 10 a + 17 + \left(13 a^{3} + 2 a^{2} + 18 a + 12\right)\cdot 19 + \left(13 a^{3} + 7 a^{2} + 12 a + 13\right)\cdot 19^{2} + \left(18 a^{3} + 6 a + 6\right)\cdot 19^{3} + \left(13 a^{3} + 2\right)\cdot 19^{4} + \left(14 a^{3} + 4 a^{2} + 12 a + 9\right)\cdot 19^{5} + \left(4 a^{3} + 6 a^{2} + 11 a + 16\right)\cdot 19^{6} + \left(4 a^{3} + 5 a^{2} + 6 a + 14\right)\cdot 19^{7} + \left(11 a^{3} + 6 a^{2} + 15 a + 6\right)\cdot 19^{8} + \left(4 a + 12\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 15 a^{3} + 12 a^{2} + 18 a + 4 + \left(17 a^{3} + 16 a^{2} + 16 a + 4\right)\cdot 19 + \left(17 a^{3} + 9 a^{2} + 4 a + 18\right)\cdot 19^{2} + \left(15 a^{3} + 5 a^{2} + 12 a + 7\right)\cdot 19^{3} + \left(8 a^{2} + 5 a + 11\right)\cdot 19^{4} + \left(12 a^{3} + 18 a^{2} + 8 a + 15\right)\cdot 19^{5} + \left(15 a^{3} + 18 a^{2} + 17 a + 4\right)\cdot 19^{6} + \left(17 a^{3} + a^{2} + 4 a + 14\right)\cdot 19^{7} + \left(10 a^{3} + 2 a^{2} + 12 a + 9\right)\cdot 19^{8} + \left(16 a^{3} + a^{2} + 13 a + 2\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 11 a^{3} + 16 a^{2} + 2 a + 13 + \left(15 a^{3} + 2 a^{2} + 10 a\right)\cdot 19 + \left(3 a^{3} + 14 a^{2} + 14 a + 1\right)\cdot 19^{2} + \left(6 a^{3} + 12 a^{2} + 2 a + 6\right)\cdot 19^{3} + \left(8 a^{3} + 16 a^{2} + 11 a + 10\right)\cdot 19^{4} + \left(4 a^{3} + 12 a^{2} + 16 a + 18\right)\cdot 19^{5} + \left(a^{3} + 18 a^{2} + 9 a + 13\right)\cdot 19^{6} + \left(14 a^{3} + 16 a^{2} + 11 a + 7\right)\cdot 19^{7} + \left(10 a^{3} + 5 a^{2} + 13 a + 2\right)\cdot 19^{8} + \left(9 a^{3} + 10 a^{2} + 6\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 4 a^{2} + a + 5 + \left(3 a^{3} + a^{2} + 16\right)\cdot 19 + \left(3 a^{3} + a^{2} + 12 a + 18\right)\cdot 19^{2} + \left(17 a^{3} + 6 a^{2} + 14 a + 9\right)\cdot 19^{3} + \left(8 a^{3} + 3 a^{2} + 4 a + 7\right)\cdot 19^{4} + \left(4 a^{3} + 16 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(17 a^{3} + 6 a^{2} + 9 a + 16\right)\cdot 19^{6} + \left(14 a^{3} + 13 a^{2} + 12 a + 10\right)\cdot 19^{7} + \left(3 a^{3} + 15 a^{2} + 2 a + 4\right)\cdot 19^{8} + \left(6 a^{3} + 12 a^{2} + 5 a + 11\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 18 a^{3} + 11 a^{2} + 7 a + 18 + \left(13 a^{3} + 9 a^{2} + 12 a\right)\cdot 19 + \left(3 a^{3} + 16 a^{2} + 5 a + 1\right)\cdot 19^{2} + \left(3 a^{3} + 14 a^{2} + 3 a + 8\right)\cdot 19^{3} + \left(3 a^{3} + 2 a^{2} + 3 a + 2\right)\cdot 19^{4} + \left(3 a^{3} + 3 a^{2} + 15 a + 8\right)\cdot 19^{5} + \left(4 a^{2} + 7 a + 10\right)\cdot 19^{6} + \left(2 a^{3} + 16 a^{2} + 10 a + 16\right)\cdot 19^{7} + \left(a^{3} + 7 a^{2} + 18 a + 7\right)\cdot 19^{8} + \left(10 a^{3} + a + 16\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.