Properties

Label 16.832...816.24t1334.a.a
Dimension $16$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $8.325\times 10^{24}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(832\!\cdots\!816\)\(\medspace = 2^{26} \cdot 3^{14} \cdot 11^{10}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.15869558403072.2
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 9.3.15869558403072.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 6x^{7} - 6x^{6} + 12x^{5} + 24x^{4} - 18x^{3} - 24x^{2} + 42x + 36 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{4} + 3x^{2} + 12x + 2 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.