Properties

Label 8.282...203.9t26.a.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $2.824\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(2824440448203\)\(\medspace = 3^{13} \cdot 11^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.2824440448203.2
Galois orbit size: $1$
Smallest permutation container: $((C_3^2:Q_8):C_3):C_2$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $C_3^2:\GL(2,3)$
Projective stem field: Galois closure of 9.3.2824440448203.2

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{7} - 9x^{6} + 3x^{5} + 18x^{4} + 15x^{3} - 9x^{2} - 27x + 6 \) 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:
$r_{ 1 }$ $=$ \( 4 a^{3} + 7 a^{2} + a + 16 + \left(4 a^{3} + 6 a^{2} + 2 a + 17\right)\cdot 19 + \left(7 a^{3} + 12 a^{2} + 3 a + 18\right)\cdot 19^{2} + \left(2 a^{3} + 9 a + 8\right)\cdot 19^{3} + \left(a^{3} + 5 a^{2} + 14 a + 6\right)\cdot 19^{4} + \left(17 a^{3} + 17 a^{2} + 7 a + 12\right)\cdot 19^{5} + \left(5 a^{3} + 18 a^{2} + 10 a + 13\right)\cdot 19^{6} + \left(17 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 19^{7} + \left(3 a^{3} + 9 a^{2} + 11 a + 4\right)\cdot 19^{8} + \left(16 a^{3} + 18 a^{2} + 15 a + 10\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{3} + 13 a^{2} + 11 a + 11 + \left(15 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 19 + \left(7 a^{3} + 13 a^{2} + 7 a + 15\right)\cdot 19^{2} + \left(9 a^{3} + 3 a^{2} + 7 a + 17\right)\cdot 19^{3} + \left(10 a^{3} + 16 a^{2} + 12 a + 18\right)\cdot 19^{4} + \left(10 a^{3} + 11 a^{2} + 4 a + 14\right)\cdot 19^{5} + \left(16 a^{3} + 15 a^{2} + 3 a + 12\right)\cdot 19^{6} + \left(14 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 19^{7} + \left(18 a^{3} + 12 a^{2} + 3 a + 15\right)\cdot 19^{8} + \left(10 a^{3} + 18 a^{2} + a + 14\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 a^{3} + 15 a^{2} + 9 a + 3 + \left(10 a^{3} + 7 a^{2} + 18 a + 3\right)\cdot 19 + \left(18 a^{3} + 10 a^{2} + 3 a + 6\right)\cdot 19^{2} + \left(18 a^{3} + 7 a^{2} + 11 a + 10\right)\cdot 19^{3} + \left(18 a^{3} + 8 a^{2} + 8 a\right)\cdot 19^{4} + \left(9 a^{3} + 4 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(12 a^{3} + 16 a^{2} + 14 a + 18\right)\cdot 19^{6} + \left(13 a^{3} + 11 a^{2} + 14 a + 15\right)\cdot 19^{7} + \left(17 a^{3} + 10 a^{2} + 10 a + 4\right)\cdot 19^{8} + \left(10 a^{3} + 8 a^{2} + 16 a + 4\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 4\cdot 19 + 7\cdot 19^{2} + 17\cdot 19^{4} + 14\cdot 19^{5} + 13\cdot 19^{6} + 7\cdot 19^{7} + 11\cdot 19^{8} + 15\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a^{3} + 2 a^{2} + a + 5 + \left(6 a^{3} + 8 a^{2} + 14 a + 9\right)\cdot 19 + \left(16 a^{3} + 6 a^{2} + 2 a + 16\right)\cdot 19^{2} + \left(10 a^{3} + 12 a^{2} + 12 a + 3\right)\cdot 19^{3} + \left(15 a^{3} + 18 a^{2} + 4 a + 8\right)\cdot 19^{4} + \left(4 a^{3} + 16 a^{2} + 9 a + 13\right)\cdot 19^{5} + \left(7 a^{3} + 8 a + 6\right)\cdot 19^{6} + \left(17 a^{3} + 5 a^{2} + 4 a + 3\right)\cdot 19^{7} + \left(7 a^{3} + 5 a + 15\right)\cdot 19^{8} + \left(11 a^{3} + 15 a^{2} + 4 a + 9\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a^{3} + 13 a^{2} + 15 a + 16 + \left(2 a^{3} + 11 a^{2} + 17 a + 17\right)\cdot 19 + \left(8 a^{3} + 13 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(17 a^{3} + 16 a^{2} + 10 a\right)\cdot 19^{3} + \left(12 a^{3} + a^{2} + 3 a + 17\right)\cdot 19^{4} + \left(13 a^{3} + 15 a^{2} + 8 a + 3\right)\cdot 19^{5} + \left(3 a^{3} + 5 a^{2} + 17 a + 6\right)\cdot 19^{6} + \left(18 a^{3} + 12 a^{2} + 14 a + 17\right)\cdot 19^{7} + \left(2 a^{3} + 9 a^{2} + 3 a + 16\right)\cdot 19^{8} + \left(6 a^{3} + 8 a^{2} + 5 a + 16\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 8 a^{3} + 3 a^{2} + 17 a + 7 + \left(7 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 19 + \left(4 a^{3} + 2 a^{2} + 4 a + 13\right)\cdot 19^{2} + \left(7 a^{3} + 7 a^{2} + 10 a + 12\right)\cdot 19^{3} + \left(7 a^{3} + 8 a^{2} + 2 a + 18\right)\cdot 19^{4} + \left(4 a^{2} + 3 a + 13\right)\cdot 19^{5} + \left(3 a^{3} + 6 a^{2} + 10 a + 5\right)\cdot 19^{6} + \left(11 a^{3} + a^{2} + 15 a + 13\right)\cdot 19^{7} + \left(16 a^{3} + 5 a^{2} + 11 a + 13\right)\cdot 19^{8} + \left(18 a^{3} + 11 a^{2} + 4 a + 10\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 6 a^{3} + 16 a + 17 + \left(17 a^{3} + 9 a^{2} + 10 a + 2\right)\cdot 19 + \left(2 a^{3} + 15 a^{2} + 9 a + 5\right)\cdot 19^{2} + \left(5 a^{3} + 17 a^{2} + 17 a\right)\cdot 19^{3} + \left(14 a^{3} + 9 a^{2} + 7 a + 3\right)\cdot 19^{4} + \left(3 a^{3} + a^{2} + 6 a + 8\right)\cdot 19^{5} + \left(11 a^{3} + 3 a^{2} + 12 a + 3\right)\cdot 19^{6} + \left(13 a^{3} + 11 a^{2} + 18 a + 16\right)\cdot 19^{7} + \left(4 a^{3} + 11 a^{2} + 5 a + 4\right)\cdot 19^{8} + \left(13 a^{2} + 15 a + 11\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 9 a^{3} + 4 a^{2} + 6 a + 3 + \left(12 a^{3} + 9 a^{2} + 14 a + 1\right)\cdot 19 + \left(10 a^{3} + 2 a^{2} + 9 a + 18\right)\cdot 19^{2} + \left(4 a^{3} + 10 a^{2} + 16 a + 1\right)\cdot 19^{3} + \left(14 a^{3} + 7 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(15 a^{3} + 4 a^{2} + 14 a + 10\right)\cdot 19^{5} + \left(15 a^{3} + 9 a^{2} + 18 a + 14\right)\cdot 19^{6} + \left(7 a^{3} + 9 a^{2} + 18 a + 9\right)\cdot 19^{7} + \left(3 a^{3} + 16 a^{2} + 3 a + 8\right)\cdot 19^{8} + \left(a^{3} + 13 a + 1\right)\cdot 19^{9} +O(19^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.