Properties

Label 8.120...776.18t157.a.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $1.202\times 10^{15}$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(1201952841091776\)\(\medspace = 2^{6} \cdot 3^{7} \cdot 97^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.127745014464.2
Galois orbit size: $1$
Smallest permutation container: 18T157
Parity: odd
Determinant: 1.291.2t1.a.a
Projective image: $C_3^2:\GL(2,3)$
Projective stem field: Galois closure of 9.3.127745014464.2

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} - 7x^{7} + 8x^{6} + 8x^{5} - x^{4} - 25x^{3} + 11x^{2} + 8x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{4} + 7x^{2} + 10x + 3 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.