Properties

Label 1.73.9t1.1c2
Dimension 1
Group $C_9$
Conductor $ 73 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_9$
Conductor:$73 $
Artin number field: Splitting field of $f= x^{9} - x^{8} - 32 x^{7} + 11 x^{6} + 278 x^{5} + 34 x^{4} - 427 x^{3} - 150 x^{2} - 8 x + 1 $ over $\Q$
Size of Galois orbit: 6
Smallest containing permutation representation: $C_9$
Parity: Even
Corresponding Dirichlet character: \(\chi_{73}(16,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{3} + x + 14 $
Roots:
$r_{ 1 }$ $=$ $ 14 a^{2} + a + 6 + \left(7 a^{2} + 2 a + 7\right)\cdot 17 + \left(12 a^{2} + 13 a + 1\right)\cdot 17^{2} + \left(14 a^{2} + 8 a + 11\right)\cdot 17^{3} + \left(10 a^{2} + 8 a + 10\right)\cdot 17^{4} + \left(10 a + 2\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 15 a^{2} + 15 a + 5 + \left(8 a^{2} + 5 a + 6\right)\cdot 17 + \left(13 a^{2} + 11 a + 13\right)\cdot 17^{2} + \left(a^{2} + 2 a + 12\right)\cdot 17^{3} + \left(11 a^{2} + 8 a + 10\right)\cdot 17^{4} + \left(11 a^{2} + a + 6\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 13 a^{2} + 7 a + 6 + \left(4 a^{2} + 9 a + 6\right)\cdot 17 + \left(a + 8\right)\cdot 17^{2} + \left(a^{2} + 15 a + 10\right)\cdot 17^{3} + \left(11 a^{2} + 12 a\right)\cdot 17^{4} + \left(13 a^{2} + 6 a + 8\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 4 a^{2} + 9 a + \left(a^{2} + 13 a + 4\right)\cdot 17 + \left(9 a^{2} + 4 a + 14\right)\cdot 17^{2} + \left(2 a + 15\right)\cdot 17^{3} + \left(14 a^{2} + 4 a + 13\right)\cdot 17^{4} + \left(8 a^{2} + 16 a + 4\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 8 + \left(a^{2} + 14 a + 8\right)\cdot 17 + \left(4 a^{2} + 4 a + 1\right)\cdot 17^{2} + \left(14 a^{2} + 2 a + 5\right)\cdot 17^{3} + \left(5 a^{2} + 15 a + 7\right)\cdot 17^{4} + \left(8 a^{2} + 16 a + 13\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 12 a^{2} + 2 a + 3 + \left(a^{2} + 6 a + 7\right)\cdot 17 + \left(12 a^{2} + 7 a + 12\right)\cdot 17^{2} + \left(16 a^{2} + 4 a + 5\right)\cdot 17^{3} + \left(3 a^{2} + 14 a\right)\cdot 17^{4} + \left(9 a^{2} + 7 a + 5\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 7 a^{2} + 11 + \left(6 a^{2} + 5 a + 4\right)\cdot 17 + \left(8 a^{2} + 15 a + 4\right)\cdot 17^{2} + \left(15 a^{2} + 9 a + 16\right)\cdot 17^{3} + \left(a^{2} + 11 a + 15\right)\cdot 17^{4} + \left(13 a^{2} + 7 a + 1\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 3 a^{2} + 11 a + 10 + \left(8 a^{2} + 7\right)\cdot 17 + \left(16 a + 10\right)\cdot 17^{2} + \left(5 a^{2} + 5 a + 4\right)\cdot 17^{3} + \left(10 a + 9\right)\cdot 17^{4} + \left(8 a^{2} + 6 a + 7\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$
$r_{ 9 }$ $=$ $ a + 3 + \left(11 a^{2} + 11 a + 16\right)\cdot 17 + \left(7 a^{2} + 10 a + 1\right)\cdot 17^{2} + \left(15 a^{2} + 16 a + 3\right)\cdot 17^{3} + \left(8 a^{2} + 16 a + 16\right)\cdot 17^{4} + \left(11 a^{2} + 10 a\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$1$
$1$$3$$(1,5,8)(2,6,7)(3,9,4)$$-\zeta_{9}^{3} - 1$
$1$$3$$(1,8,5)(2,7,6)(3,4,9)$$\zeta_{9}^{3}$
$1$$9$$(1,7,9,5,2,4,8,6,3)$$\zeta_{9}^{2}$
$1$$9$$(1,9,2,8,3,7,5,4,6)$$\zeta_{9}^{4}$
$1$$9$$(1,2,3,5,6,9,8,7,4)$$-\zeta_{9}^{5} - \zeta_{9}^{2}$
$1$$9$$(1,4,7,8,9,6,5,3,2)$$\zeta_{9}$
$1$$9$$(1,6,4,5,7,3,8,2,9)$$\zeta_{9}^{5}$
$1$$9$$(1,3,6,8,4,2,5,9,7)$$-\zeta_{9}^{4} - \zeta_{9}$
The blue line marks the conjugacy class containing complex conjugation.