Properties

 Label 1.3e3.18t1.1c3 Dimension 1 Group $C_{18}$ Conductor $3^{3}$ Root number not computed Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $1$ Group: $C_{18}$ Conductor: $27= 3^{3}$ Artin number field: Splitting field of $f= x^{18} - x^{9} + 1$ over $\Q$ Size of Galois orbit: 6 Smallest containing permutation representation: $C_{18}$ Parity: Odd Corresponding Dirichlet character: $$\chi_{27}(23,\cdot)$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{9} + 4 x^{3} + 20 x^{2} + 29 x + 28$
Roots:
 $r_{ 1 }$ $=$ $29 a^{8} + 19 a^{7} + 6 a^{6} + 23 a^{5} + 12 a^{4} + 16 a^{3} + 8 a^{2} + 8 a + 12 + \left(3 a^{8} + 17 a^{7} + 18 a^{6} + 23 a^{5} + 27 a^{4} + 17 a^{3} + 17 a^{2} + 7 a + 24\right)\cdot 31 + \left(10 a^{8} + 28 a^{7} + 29 a^{6} + 24 a^{5} + 15 a^{4} + 14 a^{3} + 30 a^{2} + 19 a + 12\right)\cdot 31^{2} + \left(13 a^{8} + 12 a^{7} + 5 a^{6} + 29 a^{5} + 11 a^{4} + 11 a^{3} + 18 a^{2} + 20 a + 2\right)\cdot 31^{3} + \left(24 a^{8} + a^{7} + 3 a^{6} + a^{5} + 27 a^{4} + 26 a^{3} + 9 a^{2} + 25 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $5 a^{8} + 9 a^{7} + 7 a^{6} + 20 a^{5} + 10 a^{4} + 3 a^{3} + 16 a^{2} + 23 a + 12 + \left(10 a^{8} + 21 a^{7} + 30 a^{6} + 25 a^{5} + 8 a^{4} + 15 a^{3} + 3 a^{2} + 23 a + 12\right)\cdot 31 + \left(13 a^{8} + 4 a^{7} + 10 a^{6} + 21 a^{5} + 12 a^{4} + 3 a^{3} + 2 a^{2} + 15 a + 1\right)\cdot 31^{2} + \left(18 a^{8} + 2 a^{7} + 4 a^{6} + 23 a^{5} + 26 a^{4} + 13 a^{3} + 12 a^{2} + 13 a\right)\cdot 31^{3} + \left(18 a^{8} + 6 a^{7} + 6 a^{6} + 30 a^{5} + 10 a^{4} + 12 a^{3} + 29 a^{2} + 3 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $5 a^{8} + 19 a^{7} + 11 a^{6} + 16 a^{5} + 18 a^{4} + 22 a^{3} + 16 a^{2} + 10 a + 6 + \left(30 a^{8} + 10 a^{7} + 30 a^{6} + 3 a^{5} + 6 a^{4} + 25 a^{3} + 7 a^{2} + 21 a + 21\right)\cdot 31 + \left(a^{8} + 7 a^{7} + 14 a^{6} + 3 a^{5} + 28 a^{4} + 10 a^{3} + a^{2} + 16 a + 8\right)\cdot 31^{2} + \left(a^{8} + 22 a^{7} + 12 a^{6} + 7 a^{5} + 30 a^{4} + 21 a^{3} + 17 a^{2} + 24\right)\cdot 31^{3} + \left(3 a^{8} + 16 a^{7} + 30 a^{6} + 24 a^{5} + 23 a^{4} + 22 a^{3} + 13 a^{2} + 27 a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $14 a^{8} + 13 a^{7} + 19 a^{6} + 21 a^{5} + 21 a^{4} + 14 a^{3} + 17 a^{2} + 15 a + 11 + \left(a^{8} + 6 a^{7} + 7 a^{6} + 9 a^{5} + 23 a^{4} + 17 a^{3} + 23 a^{2} + 7 a + 16\right)\cdot 31 + \left(10 a^{8} + 18 a^{7} + 11 a^{6} + 3 a^{5} + 18 a^{4} + 17 a^{3} + 8 a^{2} + 15 a + 24\right)\cdot 31^{2} + \left(25 a^{8} + 15 a^{7} + 8 a^{6} + 2 a^{5} + 22 a^{4} + 19 a^{3} + 15 a^{2} + 6 a + 4\right)\cdot 31^{3} + \left(6 a^{8} + 7 a^{7} + 10 a^{6} + 4 a^{5} + 9 a^{4} + 26 a^{3} + 28 a^{2} + 3 a + 20\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $15 a^{8} + a^{7} + 24 a^{6} + 14 a^{5} + 25 a^{4} + 3 a^{3} + 20 a^{2} + 5 a + 15 + \left(8 a^{8} + 24 a^{7} + 22 a^{6} + 19 a^{5} + 28 a^{4} + 26 a^{3} + 8 a^{2} + 15 a + 19\right)\cdot 31 + \left(17 a^{8} + 7 a^{7} + 26 a^{6} + 2 a^{5} + 25 a^{4} + 17 a^{3} + 10 a^{2} + 6 a + 10\right)\cdot 31^{2} + \left(2 a^{8} + 27 a^{7} + a^{6} + 2 a^{5} + 10 a^{4} + 28 a^{3} + 10 a^{2} + 14 a + 26\right)\cdot 31^{3} + \left(4 a^{8} + a^{7} + 21 a^{6} + 6 a^{5} + 26 a^{4} + 13 a^{3} + 23 a^{2} + 7 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $30 a^{8} + 4 a^{7} + 21 a^{6} + 11 a^{5} + 24 a^{4} + 13 a^{3} + 2 a^{2} + 11 a + 27 + \left(9 a^{8} + 21 a^{7} + 14 a^{6} + 29 a^{5} + 11 a^{4} + 30 a^{3} + 28 a^{2} + 3 a + 28\right)\cdot 31 + \left(28 a^{8} + 5 a^{7} + 17 a^{6} + 6 a^{5} + 3 a^{4} + 7 a^{3} + 20 a^{2} + 10 a + 3\right)\cdot 31^{2} + \left(3 a^{8} + 28 a^{6} + 29 a^{5} + 8 a^{4} + 12 a^{3} + 7 a^{2} + 22 a + 11\right)\cdot 31^{3} + \left(12 a^{8} + 7 a^{7} + 19 a^{6} + 3 a^{5} + 9 a^{4} + 5 a^{3} + 22 a^{2} + 9 a + 30\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 7 }$ $=$ $12 a^{8} + 10 a^{7} + 26 a^{6} + 17 a^{5} + 21 a^{4} + 28 a^{3} + 14 a^{2} + 14 a + 21 + \left(4 a^{8} + 23 a^{6} + 8 a^{4} + 11 a^{3} + a^{2} + 30 a + 26\right)\cdot 31 + \left(25 a^{8} + 25 a^{7} + 12 a^{6} + 21 a^{5} + 8 a^{4} + 27 a^{3} + 3 a^{2} + 24 a + 6\right)\cdot 31^{2} + \left(9 a^{8} + a^{7} + 26 a^{6} + 22 a^{5} + 9 a^{4} + 15 a^{3} + 3 a^{2} + 19 a + 24\right)\cdot 31^{3} + \left(17 a^{8} + 6 a^{7} + 18 a^{6} + 15 a^{5} + 5 a^{4} + 19 a^{3} + 25 a^{2} + 9 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 8 }$ $=$ $a^{8} + 8 a^{7} + 20 a^{6} + 4 a^{5} + 2 a^{4} + 13 a^{3} + 28 a^{2} + 17 a + 21 + \left(24 a^{8} + 25 a^{7} + 30 a^{6} + 27 a^{4} + 22 a^{3} + 2 a^{2} + 5\right)\cdot 31 + \left(29 a^{8} + 15 a^{7} + 24 a^{6} + 22 a^{5} + 3 a^{4} + 5 a^{3} + 19 a^{2} + 13 a + 26\right)\cdot 31^{2} + \left(7 a^{8} + 20 a^{7} + 30 a^{6} + 16 a^{5} + 10 a^{4} + 13 a^{3} + 26 a^{2} + 29 a + 20\right)\cdot 31^{3} + \left(7 a^{8} + 15 a^{7} + 14 a^{6} + 6 a^{5} + 16 a^{4} + 10 a^{3} + 16 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 9 }$ $=$ $a^{8} + 10 a^{7} + 27 a^{6} + 28 a^{5} + 16 a^{4} + 23 a^{3} + 28 a^{2} + 2 a + 26 + \left(28 a^{8} + 11 a^{7} + 4 a^{6} + 2 a^{5} + 18 a^{4} + 2 a^{3} + 9 a^{2} + 11 a + 5\right)\cdot 31 + \left(3 a^{8} + 2 a^{7} + 27 a^{6} + 13 a^{5} + 19 a^{4} + 8 a^{3} + 30 a^{2} + 12 a + 3\right)\cdot 31^{2} + \left(8 a^{8} + 8 a^{7} + 14 a^{6} + 11 a^{5} + 16 a^{4} + 12 a^{3} + 5 a^{2} + 8 a + 13\right)\cdot 31^{3} + \left(22 a^{8} + 8 a^{7} + 20 a^{6} + 15 a^{5} + 12 a^{4} + 5 a^{3} + 15 a^{2} + 9 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 10 }$ $=$ $9 a^{8} + 15 a^{7} + 10 a^{6} + 29 a^{5} + 29 a^{4} + 9 a^{3} + 22 a^{2} + 3 a + 27 + \left(12 a^{8} + 27 a^{7} + 29 a^{6} + 19 a^{5} + 28 a^{4} + 9 a^{3} + 23 a^{2} + 24 a + 26\right)\cdot 31 + \left(17 a^{8} + 15 a^{7} + 20 a^{6} + 22 a^{5} + 12 a^{4} + 27 a^{3} + 15 a^{2} + a + 10\right)\cdot 31^{2} + \left(25 a^{8} + 13 a^{7} + 15 a^{6} + 5 a^{5} + 12 a^{4} + 4 a^{3} + 14 a^{2} + 6 a + 15\right)\cdot 31^{3} + \left(29 a^{8} + 6 a^{7} + 16 a^{6} + 2 a^{4} + 24 a^{3} + 26 a^{2} + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 11 }$ $=$ $3 a^{8} + 25 a^{7} + 11 a^{6} + 9 a^{5} + 5 a^{4} + 13 a^{3} + 4 a^{2} + a + 3 + \left(18 a^{8} + 27 a^{7} + 23 a^{6} + 28 a^{5} + 4 a^{4} + 18 a^{3} + 9 a^{2} + 25 a + 14\right)\cdot 31 + \left(6 a^{8} + 26 a^{7} + 29 a^{6} + 26 a^{5} + 19 a^{4} + 13 a^{3} + 19 a^{2} + 16\right)\cdot 31^{2} + \left(8 a^{8} + 10 a^{7} + 22 a^{6} + 22 a^{5} + 13 a^{4} + 13 a^{3} + 15 a^{2} + 9 a + 4\right)\cdot 31^{3} + \left(24 a^{8} + 16 a^{7} + 17 a^{6} + 21 a^{5} + 30 a^{4} + 3 a^{3} + 9 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 12 }$ $=$ $6 a^{8} + 7 a^{7} + 29 a^{6} + 27 a^{5} + 11 a^{4} + 15 a^{3} + 19 a^{2} + 27 a + 24 + \left(16 a^{8} + 3 a^{7} + 20 a^{6} + 10 a^{5} + 27 a^{4} + 7 a^{3} + 20 a^{2} + 11 a + 6\right)\cdot 31 + \left(24 a^{8} + 8 a^{7} + 22 a^{6} + 27 a^{5} + 13 a^{4} + 20 a^{3} + 22 a^{2} + 9\right)\cdot 31^{2} + \left(30 a^{8} + 24 a^{6} + 19 a^{5} + 18 a^{4} + 3 a^{3} + 25 a^{2} + 21 a + 23\right)\cdot 31^{3} + \left(29 a^{8} + 12 a^{7} + 22 a^{6} + 21 a^{5} + 29 a^{4} + 26 a^{3} + 19 a^{2} + 29 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 13 }$ $=$ $21 a^{8} + 2 a^{7} + 30 a^{6} + 22 a^{5} + 29 a^{4} + 18 a^{3} + 9 a^{2} + 9 a + 29 + \left(22 a^{8} + 13 a^{7} + 19 a^{6} + 6 a^{5} + 25 a^{4} + a^{3} + 12 a^{2} + 24 a + 10\right)\cdot 31 + \left(26 a^{8} + 8 a^{7} + 19 a^{6} + 16 a^{5} + 6 a^{4} + 20 a^{3} + 28 a^{2} + 17 a + 11\right)\cdot 31^{2} + \left(7 a^{8} + 16 a^{7} + 29 a^{6} + 9 a^{5} + 10 a^{4} + 3 a^{3} + 8 a^{2} + 21 a + 4\right)\cdot 31^{3} + \left(20 a^{8} + 23 a^{7} + 8 a^{6} + 13 a^{5} + 29 a^{4} + 16 a^{3} + 27 a^{2} + 26 a + 30\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 14 }$ $=$ $25 a^{8} + 14 a^{7} + 4 a^{6} + 7 a^{5} + 19 a^{4} + 15 a^{3} + 18 a^{2} + 22 a + 29 + \left(27 a^{8} + 15 a^{7} + a^{6} + 5 a^{5} + 26 a^{4} + 24 a^{3} + 24 a^{2} + 6 a + 12\right)\cdot 31 + \left(18 a^{8} + 10 a^{7} + 26 a^{6} + 18 a^{5} + 14 a^{4} + 21 a^{3} + 9 a^{2} + 2 a + 3\right)\cdot 31^{2} + \left(4 a^{8} + 8 a^{7} + 26 a^{6} + 21 a^{5} + 25 a^{4} + 4 a^{3} + 23 a^{2} + 19 a + 10\right)\cdot 31^{3} + \left(5 a^{8} + 9 a^{7} + 9 a^{6} + 24 a^{5} + 3 a^{4} + 8 a^{3} + 10 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 15 }$ $=$ $25 a^{8} + 2 a^{7} + 24 a^{6} + 18 a^{5} + 28 a^{4} + 17 a^{3} + 18 a^{2} + 19 a + 30 + \left(3 a^{8} + 9 a^{7} + 26 a^{6} + 24 a^{5} + 5 a^{4} + 2 a^{3} + 13 a^{2} + 29 a + 3\right)\cdot 31 + \left(25 a^{8} + 21 a^{7} + 19 a^{6} + 14 a^{5} + 14 a^{4} + 12 a^{3} + 30 a^{2} + a + 19\right)\cdot 31^{2} + \left(21 a^{8} + 3 a^{6} + 12 a^{5} + 14 a^{4} + 28 a^{3} + 7 a^{2} + 22 a + 24\right)\cdot 31^{3} + \left(5 a^{8} + 6 a^{7} + 11 a^{6} + 22 a^{5} + 25 a^{4} + 2 a^{3} + 2 a^{2} + 25 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 16 }$ $=$ $8 a^{8} + 3 a^{7} + 2 a^{6} + 12 a^{5} + 12 a^{4} + 8 a^{3} + 23 a^{2} + 13 a + 24 + \left(17 a^{8} + 28 a^{7} + 25 a^{6} + a^{5} + 9 a^{4} + 4 a^{3} + 14 a^{2} + 30 a + 18\right)\cdot 31 + \left(3 a^{8} + 27 a^{7} + 29 a^{6} + 5 a^{5} + 30 a^{4} + 17 a^{3} + 6 a^{2} + 13 a + 26\right)\cdot 31^{2} + \left(11 a^{8} + a^{7} + 6 a^{6} + 23 a^{5} + 26 a^{4} + 6 a^{3} + a^{2} + 18 a + 10\right)\cdot 31^{3} + \left(25 a^{8} + 17 a^{7} + 4 a^{6} + 26 a^{5} + 18 a^{4} + 11 a^{3} + 7 a^{2} + 27 a\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 17 }$ $=$ $13 a^{8} + 5 a^{7} + 27 a^{6} + 8 a^{5} + a^{4} + 15 a^{3} + 7 a^{2} + 25 a + 13 + \left(4 a^{8} + 10 a^{7} + 15 a^{6} + 14 a^{5} + 29 a^{4} + 17 a^{3} + 13 a^{2} + 21 a + 28\right)\cdot 31 + \left(7 a^{8} + 27 a^{7} + 5 a^{6} + a^{5} + 16 a^{4} + 30 a^{3} + a^{2} + 23 a + 3\right)\cdot 31^{2} + \left(20 a^{8} + 23 a^{7} + 6 a^{6} + 6 a^{5} + 6 a^{4} + 19 a^{3} + 5 a^{2} + 7 a\right)\cdot 31^{3} + \left(2 a^{8} + 12 a^{7} + 23 a^{6} + 3 a^{5} + 5 a^{4} + 13 a^{3} + 7 a^{2} + 14 a + 12\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 18 }$ $=$ $26 a^{8} + 20 a^{7} + 12 a^{6} + 24 a^{5} + 27 a^{4} + 3 a^{3} + 10 a^{2} + 24 a + 11 + \left(4 a^{8} + 6 a^{7} + 26 a^{6} + 21 a^{5} + 22 a^{4} + 24 a^{3} + 13 a^{2} + 15 a + 26\right)\cdot 31 + \left(9 a^{8} + 17 a^{7} + 21 a^{6} + 27 a^{5} + 13 a^{4} + 2 a^{3} + 18 a^{2} + 20 a + 17\right)\cdot 31^{2} + \left(27 a^{8} + 30 a^{7} + 8 a^{6} + 12 a^{5} + 4 a^{4} + 15 a^{3} + 28 a^{2} + 18 a + 27\right)\cdot 31^{3} + \left(19 a^{8} + 11 a^{7} + 19 a^{6} + 5 a^{5} + 23 a^{4} + 30 a^{3} + 19 a^{2} + 22 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 18 }$

 Cycle notation $(1,18,7,12,13,6)(2,5,14,11,8,17)(3,16,9,10,15,4)$ $(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)$ $(1,15,5,7,3,11,13,9,17)(2,18,4,14,12,16,8,6,10)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 18 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,18)(14,17)(15,16)$ $-1$ $1$ $3$ $(1,7,13)(2,14,8)(3,9,15)(4,16,10)(5,11,17)(6,18,12)$ $\zeta_{9}^{3}$ $1$ $3$ $(1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)$ $-\zeta_{9}^{3} - 1$ $1$ $6$ $(1,18,7,12,13,6)(2,5,14,11,8,17)(3,16,9,10,15,4)$ $\zeta_{9}^{3} + 1$ $1$ $6$ $(1,6,13,12,7,18)(2,17,8,11,14,5)(3,4,15,10,9,16)$ $-\zeta_{9}^{3}$ $1$ $9$ $(1,15,5,7,3,11,13,9,17)(2,18,4,14,12,16,8,6,10)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,5,3,13,17,15,7,11,9)(2,4,12,8,10,18,14,16,6)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $1$ $9$ $(1,3,17,7,9,5,13,15,11)(2,12,10,14,6,4,8,18,16)$ $-\zeta_{9}^{4} - \zeta_{9}$ $1$ $9$ $(1,11,15,13,5,9,7,17,3)(2,16,18,8,4,6,14,10,12)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,9,11,7,15,17,13,3,5)(2,6,16,14,18,10,8,12,4)$ $\zeta_{9}$ $1$ $9$ $(1,17,9,13,11,3,7,5,15)(2,10,6,8,16,12,14,4,18)$ $\zeta_{9}^{5}$ $1$ $18$ $(1,4,11,6,15,14,13,10,5,12,9,2,7,16,17,18,3,8)$ $-\zeta_{9}$ $1$ $18$ $(1,14,9,18,11,10,7,8,15,12,17,4,13,2,3,6,5,16)$ $-\zeta_{9}^{5}$ $1$ $18$ $(1,10,17,6,9,8,13,16,11,12,3,14,7,4,5,18,15,2)$ $\zeta_{9}^{4} + \zeta_{9}$ $1$ $18$ $(1,2,15,18,5,4,7,14,3,12,11,16,13,8,9,6,17,10)$ $-\zeta_{9}^{2}$ $1$ $18$ $(1,16,5,6,3,2,13,4,17,12,15,8,7,10,11,18,9,14)$ $-\zeta_{9}^{4}$ $1$ $18$ $(1,8,3,18,17,16,7,2,9,12,5,10,13,14,15,6,11,4)$ $\zeta_{9}^{5} + \zeta_{9}^{2}$
The blue line marks the conjugacy class containing complex conjugation.