Properties

 Label 16.303...929.24t1334.a.a Dimension $16$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $3.036\times 10^{24}$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $16$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$303\!\cdots\!929$$$$\medspace = 3^{18} \cdot 97^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.17964142659.1 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 9.3.17964142659.1

Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{8} + 7x^{6} - 3x^{5} - 12x^{4} + 8x^{3} + 3x - 2$$ x^9 - 3*x^8 + 7*x^6 - 3*x^5 - 12*x^4 + 8*x^3 + 3*x - 2 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{4} + 3x^{2} + 19x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$8 a^{3} + 20 a^{2} + 6 a + 6 + \left(17 a^{3} + 2 a^{2} + 14 a + 8\right)\cdot 23 + \left(3 a^{3} + 18 a^{2} + 19 a\right)\cdot 23^{2} + \left(18 a^{3} + a^{2} + a + 6\right)\cdot 23^{3} + \left(13 a^{3} + 4 a^{2} + 15 a + 2\right)\cdot 23^{4} + \left(11 a^{3} + 15 a^{2} + 12 a + 20\right)\cdot 23^{5} + \left(a^{3} + 14 a^{2} + 7 a + 3\right)\cdot 23^{6} + \left(7 a^{3} + 22 a^{2} + 16 a + 17\right)\cdot 23^{7} + \left(21 a^{3} + 21 a^{2} + 18 a + 21\right)\cdot 23^{8} + \left(21 a^{3} + 12 a^{2} + 16 a + 2\right)\cdot 23^{9} +O(23^{10})$$ 8*a^3 + 20*a^2 + 6*a + 6 + (17*a^3 + 2*a^2 + 14*a + 8)*23 + (3*a^3 + 18*a^2 + 19*a)*23^2 + (18*a^3 + a^2 + a + 6)*23^3 + (13*a^3 + 4*a^2 + 15*a + 2)*23^4 + (11*a^3 + 15*a^2 + 12*a + 20)*23^5 + (a^3 + 14*a^2 + 7*a + 3)*23^6 + (7*a^3 + 22*a^2 + 16*a + 17)*23^7 + (21*a^3 + 21*a^2 + 18*a + 21)*23^8 + (21*a^3 + 12*a^2 + 16*a + 2)*23^9+O(23^10) $r_{ 2 }$ $=$ $$10 a^{3} + 12 a^{2} + 22 a + 7 + \left(21 a^{3} + 16 a^{2} + 11 a + 4\right)\cdot 23 + \left(14 a^{3} + 6 a^{2} + 9 a + 21\right)\cdot 23^{2} + \left(20 a^{3} + 9 a^{2} + 18 a + 1\right)\cdot 23^{3} + \left(5 a^{3} + 17 a^{2} + 12 a + 20\right)\cdot 23^{4} + \left(7 a^{3} + 18 a^{2} + 15 a + 12\right)\cdot 23^{5} + \left(18 a^{3} + 5 a^{2} + 15 a + 5\right)\cdot 23^{6} + \left(22 a^{3} + 7 a^{2} + 11 a + 4\right)\cdot 23^{7} + \left(22 a^{3} + 14 a^{2} + 4 a + 4\right)\cdot 23^{8} + \left(22 a^{3} + a^{2} + 18 a + 9\right)\cdot 23^{9} +O(23^{10})$$ 10*a^3 + 12*a^2 + 22*a + 7 + (21*a^3 + 16*a^2 + 11*a + 4)*23 + (14*a^3 + 6*a^2 + 9*a + 21)*23^2 + (20*a^3 + 9*a^2 + 18*a + 1)*23^3 + (5*a^3 + 17*a^2 + 12*a + 20)*23^4 + (7*a^3 + 18*a^2 + 15*a + 12)*23^5 + (18*a^3 + 5*a^2 + 15*a + 5)*23^6 + (22*a^3 + 7*a^2 + 11*a + 4)*23^7 + (22*a^3 + 14*a^2 + 4*a + 4)*23^8 + (22*a^3 + a^2 + 18*a + 9)*23^9+O(23^10) $r_{ 3 }$ $=$ $$17 a^{3} + 19 a^{2} + 9 a + 8 + \left(a^{3} + 21 a^{2} + a + 13\right)\cdot 23 + \left(16 a^{3} + 21 a^{2} + 22 a + 8\right)\cdot 23^{2} + \left(2 a^{3} + 11 a^{2} + 7 a + 20\right)\cdot 23^{3} + \left(9 a^{3} + 11 a^{2} + 5 a + 16\right)\cdot 23^{4} + \left(a^{3} + 11 a^{2} + 16 a + 4\right)\cdot 23^{5} + \left(12 a^{3} + 3 a^{2} + 7 a + 22\right)\cdot 23^{6} + \left(18 a^{3} + 19 a^{2} + 7 a + 12\right)\cdot 23^{7} + \left(2 a^{3} + 18 a^{2} + 6 a + 16\right)\cdot 23^{8} + \left(18 a^{3} + 19 a^{2} + 3 a + 18\right)\cdot 23^{9} +O(23^{10})$$ 17*a^3 + 19*a^2 + 9*a + 8 + (a^3 + 21*a^2 + a + 13)*23 + (16*a^3 + 21*a^2 + 22*a + 8)*23^2 + (2*a^3 + 11*a^2 + 7*a + 20)*23^3 + (9*a^3 + 11*a^2 + 5*a + 16)*23^4 + (a^3 + 11*a^2 + 16*a + 4)*23^5 + (12*a^3 + 3*a^2 + 7*a + 22)*23^6 + (18*a^3 + 19*a^2 + 7*a + 12)*23^7 + (2*a^3 + 18*a^2 + 6*a + 16)*23^8 + (18*a^3 + 19*a^2 + 3*a + 18)*23^9+O(23^10) $r_{ 4 }$ $=$ $$9 a^{3} + 3 a^{2} + 16 a + 12 + \left(15 a^{3} + 19 a^{2} + 5 a + 10\right)\cdot 23 + \left(15 a^{3} + 7 a^{2} + 7 a + 22\right)\cdot 23^{2} + \left(14 a^{3} + 4 a^{2} + 17 a + 11\right)\cdot 23^{3} + \left(10 a^{3} + 2 a^{2} + 3 a\right)\cdot 23^{4} + \left(6 a^{3} + 12 a^{2} + 15 a + 17\right)\cdot 23^{5} + \left(13 a^{3} + 6 a^{2} + 22 a + 15\right)\cdot 23^{6} + \left(16 a^{3} + 10 a^{2} + a + 7\right)\cdot 23^{7} + \left(10 a^{3} + 2 a^{2} + 3 a + 8\right)\cdot 23^{8} + \left(18 a^{3} + 3 a^{2} + 12 a + 19\right)\cdot 23^{9} +O(23^{10})$$ 9*a^3 + 3*a^2 + 16*a + 12 + (15*a^3 + 19*a^2 + 5*a + 10)*23 + (15*a^3 + 7*a^2 + 7*a + 22)*23^2 + (14*a^3 + 4*a^2 + 17*a + 11)*23^3 + (10*a^3 + 2*a^2 + 3*a)*23^4 + (6*a^3 + 12*a^2 + 15*a + 17)*23^5 + (13*a^3 + 6*a^2 + 22*a + 15)*23^6 + (16*a^3 + 10*a^2 + a + 7)*23^7 + (10*a^3 + 2*a^2 + 3*a + 8)*23^8 + (18*a^3 + 3*a^2 + 12*a + 19)*23^9+O(23^10) $r_{ 5 }$ $=$ $$19 a^{3} + 11 a^{2} + 2 a + 13 + \left(2 a^{3} + 17 a^{2} + 6 a + 16\right)\cdot 23 + \left(19 a^{3} + 14 a^{2} + 21 a + 6\right)\cdot 23^{2} + \left(6 a^{3} + 19 a^{2} + 9 a + 10\right)\cdot 23^{3} + \left(22 a^{3} + a^{2} + 22 a\right)\cdot 23^{4} + \left(a^{3} + 9 a^{2} + 14 a + 15\right)\cdot 23^{5} + \left(4 a^{3} + 9 a^{2} + 17 a + 3\right)\cdot 23^{6} + \left(14 a^{3} + 16 a^{2} + 16 a + 10\right)\cdot 23^{7} + \left(18 a^{3} + 21 a^{2} + 9 a + 10\right)\cdot 23^{8} + \left(10 a^{3} + 12 a^{2} + 3 a + 13\right)\cdot 23^{9} +O(23^{10})$$ 19*a^3 + 11*a^2 + 2*a + 13 + (2*a^3 + 17*a^2 + 6*a + 16)*23 + (19*a^3 + 14*a^2 + 21*a + 6)*23^2 + (6*a^3 + 19*a^2 + 9*a + 10)*23^3 + (22*a^3 + a^2 + 22*a)*23^4 + (a^3 + 9*a^2 + 14*a + 15)*23^5 + (4*a^3 + 9*a^2 + 17*a + 3)*23^6 + (14*a^3 + 16*a^2 + 16*a + 10)*23^7 + (18*a^3 + 21*a^2 + 9*a + 10)*23^8 + (10*a^3 + 12*a^2 + 3*a + 13)*23^9+O(23^10) $r_{ 6 }$ $=$ $$4 a^{3} + 4 a^{2} + 18 a + 17 + \left(20 a^{3} + 2 a^{2} + 7 a + 18\right)\cdot 23 + \left(22 a^{3} + 22 a^{2} + 22 a + 2\right)\cdot 23^{2} + \left(14 a^{3} + 16 a^{2} + 6 a + 18\right)\cdot 23^{3} + \left(18 a^{3} + 12 a^{2} + 21 a + 9\right)\cdot 23^{4} + \left(21 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 23^{5} + \left(8 a^{3} + 11 a^{2} + 4 a + 1\right)\cdot 23^{6} + \left(4 a^{3} + 16 a^{2} + 10 a + 10\right)\cdot 23^{7} + \left(4 a^{3} + 6 a^{2} + a + 19\right)\cdot 23^{8} + \left(11 a^{3} + 6 a^{2} + 16 a\right)\cdot 23^{9} +O(23^{10})$$ 4*a^3 + 4*a^2 + 18*a + 17 + (20*a^3 + 2*a^2 + 7*a + 18)*23 + (22*a^3 + 22*a^2 + 22*a + 2)*23^2 + (14*a^3 + 16*a^2 + 6*a + 18)*23^3 + (18*a^3 + 12*a^2 + 21*a + 9)*23^4 + (21*a^3 + 7*a^2 + 2*a + 10)*23^5 + (8*a^3 + 11*a^2 + 4*a + 1)*23^6 + (4*a^3 + 16*a^2 + 10*a + 10)*23^7 + (4*a^3 + 6*a^2 + a + 19)*23^8 + (11*a^3 + 6*a^2 + 16*a)*23^9+O(23^10) $r_{ 7 }$ $=$ $$2 a^{3} + 19 a^{2} + 6 a + 11 + \left(16 a^{3} + 21 a^{2} + 18 a + 1\right)\cdot 23 + \left(3 a^{3} + 20 a^{2} + 19 a + 21\right)\cdot 23^{2} + \left(21 a^{3} + 22 a^{2} + 19 a + 22\right)\cdot 23^{3} + \left(2 a^{3} + 3 a^{2} + 5 a + 13\right)\cdot 23^{4} + \left(6 a^{3} + 11 a^{2} + 15 a + 16\right)\cdot 23^{5} + \left(22 a^{3} + 13 a^{2} + 11 a + 4\right)\cdot 23^{6} + \left(17 a^{3} + 19 a^{2} + 17 a + 1\right)\cdot 23^{7} + \left(9 a^{3} + 14 a^{2} + 22 a + 2\right)\cdot 23^{8} + \left(17 a^{3} + 18\right)\cdot 23^{9} +O(23^{10})$$ 2*a^3 + 19*a^2 + 6*a + 11 + (16*a^3 + 21*a^2 + 18*a + 1)*23 + (3*a^3 + 20*a^2 + 19*a + 21)*23^2 + (21*a^3 + 22*a^2 + 19*a + 22)*23^3 + (2*a^3 + 3*a^2 + 5*a + 13)*23^4 + (6*a^3 + 11*a^2 + 15*a + 16)*23^5 + (22*a^3 + 13*a^2 + 11*a + 4)*23^6 + (17*a^3 + 19*a^2 + 17*a + 1)*23^7 + (9*a^3 + 14*a^2 + 22*a + 2)*23^8 + (17*a^3 + 18)*23^9+O(23^10) $r_{ 8 }$ $=$ $$19 + 22\cdot 23 + 16\cdot 23^{3} + 11\cdot 23^{4} + 19\cdot 23^{5} + 18\cdot 23^{6} + 23^{7} + 17\cdot 23^{8} +O(23^{10})$$ 19 + 22*23 + 16*23^3 + 11*23^4 + 19*23^5 + 18*23^6 + 23^7 + 17*23^8+O(23^10) $r_{ 9 }$ $=$ $$4 a^{2} + 13 a + 2 + \left(20 a^{3} + 13 a^{2} + 3 a + 19\right)\cdot 23 + \left(18 a^{3} + 2 a^{2} + 16 a + 7\right)\cdot 23^{2} + \left(15 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 23^{3} + \left(8 a^{3} + 15 a^{2} + 5 a + 16\right)\cdot 23^{4} + \left(12 a^{3} + 6 a^{2} + 22 a + 21\right)\cdot 23^{5} + \left(11 a^{3} + 4 a^{2} + 4 a + 15\right)\cdot 23^{6} + \left(13 a^{3} + 3 a^{2} + 10 a + 3\right)\cdot 23^{7} + \left(a^{3} + 14 a^{2} + 2 a + 15\right)\cdot 23^{8} + \left(17 a^{3} + 11 a^{2} + 21 a + 8\right)\cdot 23^{9} +O(23^{10})$$ 4*a^2 + 13*a + 2 + (20*a^3 + 13*a^2 + 3*a + 19)*23 + (18*a^3 + 2*a^2 + 16*a + 7)*23^2 + (15*a^3 + 5*a^2 + 9*a + 7)*23^3 + (8*a^3 + 15*a^2 + 5*a + 16)*23^4 + (12*a^3 + 6*a^2 + 22*a + 21)*23^5 + (11*a^3 + 4*a^2 + 4*a + 15)*23^6 + (13*a^3 + 3*a^2 + 10*a + 3)*23^7 + (a^3 + 14*a^2 + 2*a + 15)*23^8 + (17*a^3 + 11*a^2 + 21*a + 8)*23^9+O(23^10)

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.