# Properties

 Label 12.731...449.18t218.a.a Dimension $12$ Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor $7.319\times 10^{15}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor: $$7319366412146449$$$$\medspace = 1489^{5}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.5.4915625528641.1 Galois orbit size: $1$ Smallest permutation container: 18T218 Parity: even Determinant: 1.1489.2t1.a.a Projective image: $C_3^3.S_4$ Projective stem field: Galois closure of 9.5.4915625528641.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 2x^{8} - x^{7} + 11x^{6} - 25x^{5} - 33x^{4} + 35x^{3} + 18x^{2} + 14x + 29$$ x^9 - 2*x^8 - x^7 + 11*x^6 - 25*x^5 - 33*x^4 + 35*x^3 + 18*x^2 + 14*x + 29 .

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $$x^{3} + 7x + 59$$

Roots:
 $r_{ 1 }$ $=$ $$6 a^{2} + 7 a + 24 + \left(6 a^{2} + 47 a + 51\right)\cdot 61 + \left(31 a^{2} + 30 a\right)\cdot 61^{2} + \left(32 a^{2} + 28 a + 53\right)\cdot 61^{3} + \left(43 a^{2} + 40 a + 11\right)\cdot 61^{4} + \left(25 a^{2} + 4 a + 38\right)\cdot 61^{5} + \left(43 a^{2} + 17 a + 56\right)\cdot 61^{6} + \left(16 a^{2} + a + 60\right)\cdot 61^{7} + \left(50 a^{2} + 7 a + 11\right)\cdot 61^{8} + \left(59 a^{2} + 34 a + 3\right)\cdot 61^{9} +O(61^{10})$$ 6*a^2 + 7*a + 24 + (6*a^2 + 47*a + 51)*61 + (31*a^2 + 30*a)*61^2 + (32*a^2 + 28*a + 53)*61^3 + (43*a^2 + 40*a + 11)*61^4 + (25*a^2 + 4*a + 38)*61^5 + (43*a^2 + 17*a + 56)*61^6 + (16*a^2 + a + 60)*61^7 + (50*a^2 + 7*a + 11)*61^8 + (59*a^2 + 34*a + 3)*61^9+O(61^10) $r_{ 2 }$ $=$ $$56 a^{2} + 33 a + 33 + \left(6 a^{2} + 36 a + 20\right)\cdot 61 + \left(37 a^{2} + 5 a + 22\right)\cdot 61^{2} + \left(47 a^{2} + 51 a + 59\right)\cdot 61^{3} + \left(44 a^{2} + 14 a + 44\right)\cdot 61^{4} + \left(50 a^{2} + 48 a + 46\right)\cdot 61^{5} + \left(19 a^{2} + 16 a + 57\right)\cdot 61^{6} + \left(33 a^{2} + a + 24\right)\cdot 61^{7} + \left(a^{2} + 29 a + 33\right)\cdot 61^{8} + \left(9 a^{2} + 31 a + 21\right)\cdot 61^{9} +O(61^{10})$$ 56*a^2 + 33*a + 33 + (6*a^2 + 36*a + 20)*61 + (37*a^2 + 5*a + 22)*61^2 + (47*a^2 + 51*a + 59)*61^3 + (44*a^2 + 14*a + 44)*61^4 + (50*a^2 + 48*a + 46)*61^5 + (19*a^2 + 16*a + 57)*61^6 + (33*a^2 + a + 24)*61^7 + (a^2 + 29*a + 33)*61^8 + (9*a^2 + 31*a + 21)*61^9+O(61^10) $r_{ 3 }$ $=$ $$24 a^{2} + 48 a + 47 + \left(47 a^{2} + a + 40\right)\cdot 61 + \left(37 a^{2} + 8 a + 11\right)\cdot 61^{2} + \left(27 a^{2} + 53 a + 30\right)\cdot 61^{3} + \left(46 a^{2} + 2 a + 25\right)\cdot 61^{4} + \left(59 a^{2} + 50 a + 34\right)\cdot 61^{5} + \left(39 a^{2} + 6 a + 40\right)\cdot 61^{6} + \left(7 a^{2} + 37 a + 18\right)\cdot 61^{7} + \left(26 a^{2} + 3 a + 21\right)\cdot 61^{8} + \left(7 a^{2} + 22 a + 43\right)\cdot 61^{9} +O(61^{10})$$ 24*a^2 + 48*a + 47 + (47*a^2 + a + 40)*61 + (37*a^2 + 8*a + 11)*61^2 + (27*a^2 + 53*a + 30)*61^3 + (46*a^2 + 2*a + 25)*61^4 + (59*a^2 + 50*a + 34)*61^5 + (39*a^2 + 6*a + 40)*61^6 + (7*a^2 + 37*a + 18)*61^7 + (26*a^2 + 3*a + 21)*61^8 + (7*a^2 + 22*a + 43)*61^9+O(61^10) $r_{ 4 }$ $=$ $$47 a^{2} + 54 a + 52 + \left(7 a^{2} + 32 a + 44\right)\cdot 61 + \left(36 a^{2} + 43 a + 17\right)\cdot 61^{2} + \left(41 a^{2} + 22 a + 31\right)\cdot 61^{3} + \left(42 a^{2} + 54 a + 55\right)\cdot 61^{4} + \left(38 a^{2} + 5 a + 10\right)\cdot 61^{5} + \left(23 a^{2} + 46 a + 55\right)\cdot 61^{6} + \left(34 a^{2} + 27 a + 29\right)\cdot 61^{7} + \left(60 a^{2} + 15 a + 44\right)\cdot 61^{8} + \left(2 a^{2} + 29 a + 13\right)\cdot 61^{9} +O(61^{10})$$ 47*a^2 + 54*a + 52 + (7*a^2 + 32*a + 44)*61 + (36*a^2 + 43*a + 17)*61^2 + (41*a^2 + 22*a + 31)*61^3 + (42*a^2 + 54*a + 55)*61^4 + (38*a^2 + 5*a + 10)*61^5 + (23*a^2 + 46*a + 55)*61^6 + (34*a^2 + 27*a + 29)*61^7 + (60*a^2 + 15*a + 44)*61^8 + (2*a^2 + 29*a + 13)*61^9+O(61^10) $r_{ 5 }$ $=$ $$19 a^{2} + 35 a + 23 + \left(46 a^{2} + 52 a + 21\right)\cdot 61 + \left(48 a^{2} + 11 a + 56\right)\cdot 61^{2} + \left(32 a^{2} + 48 a + 30\right)\cdot 61^{3} + \left(34 a^{2} + 52 a + 17\right)\cdot 61^{4} + \left(32 a^{2} + 6 a + 43\right)\cdot 61^{5} + \left(17 a^{2} + 59 a + 26\right)\cdot 61^{6} + \left(54 a^{2} + 31 a + 41\right)\cdot 61^{7} + \left(59 a^{2} + 16 a\right)\cdot 61^{8} + \left(48 a^{2} + 25\right)\cdot 61^{9} +O(61^{10})$$ 19*a^2 + 35*a + 23 + (46*a^2 + 52*a + 21)*61 + (48*a^2 + 11*a + 56)*61^2 + (32*a^2 + 48*a + 30)*61^3 + (34*a^2 + 52*a + 17)*61^4 + (32*a^2 + 6*a + 43)*61^5 + (17*a^2 + 59*a + 26)*61^6 + (54*a^2 + 31*a + 41)*61^7 + (59*a^2 + 16*a)*61^8 + (48*a^2 + 25)*61^9+O(61^10) $r_{ 6 }$ $=$ $$29 + 5\cdot 61 + 47\cdot 61^{2} + 60\cdot 61^{3} + 20\cdot 61^{4} + 48\cdot 61^{5} + 42\cdot 61^{6} + 55\cdot 61^{7} + 27\cdot 61^{8} + 2\cdot 61^{9} +O(61^{10})$$ 29 + 5*61 + 47*61^2 + 60*61^3 + 20*61^4 + 48*61^5 + 42*61^6 + 55*61^7 + 27*61^8 + 2*61^9+O(61^10) $r_{ 7 }$ $=$ $$39 + 58\cdot 61 + 41\cdot 61^{2} + 20\cdot 61^{3} + 37\cdot 61^{4} + 50\cdot 61^{5} + 3\cdot 61^{6} + 50\cdot 61^{7} + 17\cdot 61^{8} + 56\cdot 61^{9} +O(61^{10})$$ 39 + 58*61 + 41*61^2 + 20*61^3 + 37*61^4 + 50*61^5 + 3*61^6 + 50*61^7 + 17*61^8 + 56*61^9+O(61^10) $r_{ 8 }$ $=$ $$31 a^{2} + 6 a + 39 + \left(7 a^{2} + 12 a + 37\right)\cdot 61 + \left(53 a^{2} + 22 a + 42\right)\cdot 61^{2} + \left(40 a + 47\right)\cdot 61^{3} + \left(32 a^{2} + 17 a + 59\right)\cdot 61^{4} + \left(36 a^{2} + 6 a + 47\right)\cdot 61^{5} + \left(38 a^{2} + 37 a + 13\right)\cdot 61^{6} + \left(36 a^{2} + 22 a + 52\right)\cdot 61^{7} + \left(45 a^{2} + 50 a + 30\right)\cdot 61^{8} + \left(54 a^{2} + 4 a + 40\right)\cdot 61^{9} +O(61^{10})$$ 31*a^2 + 6*a + 39 + (7*a^2 + 12*a + 37)*61 + (53*a^2 + 22*a + 42)*61^2 + (40*a + 47)*61^3 + (32*a^2 + 17*a + 59)*61^4 + (36*a^2 + 6*a + 47)*61^5 + (38*a^2 + 37*a + 13)*61^6 + (36*a^2 + 22*a + 52)*61^7 + (45*a^2 + 50*a + 30)*61^8 + (54*a^2 + 4*a + 40)*61^9+O(61^10) $r_{ 9 }$ $=$ $$21 + 24\cdot 61 + 3\cdot 61^{2} + 32\cdot 61^{3} + 31\cdot 61^{4} + 45\cdot 61^{5} + 7\cdot 61^{6} + 32\cdot 61^{7} + 55\cdot 61^{8} + 37\cdot 61^{9} +O(61^{10})$$ 21 + 24*61 + 3*61^2 + 32*61^3 + 31*61^4 + 45*61^5 + 7*61^6 + 32*61^7 + 55*61^8 + 37*61^9+O(61^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $12$ $27$ $2$ $(1,3)(6,7)$ $0$ $54$ $2$ $(1,6)(2,4)(3,7)(8,9)$ $2$ $6$ $3$ $(6,7,9)$ $0$ $8$ $3$ $(1,3,8)(2,4,5)(6,7,9)$ $3$ $12$ $3$ $(1,3,8)(6,7,9)$ $-3$ $72$ $3$ $(1,6,2)(3,7,4)(5,8,9)$ $0$ $54$ $4$ $(1,6,3,7)(8,9)$ $0$ $54$ $6$ $(1,3)(2,4)(6,9,7)$ $0$ $108$ $6$ $(1,6,8,9,3,7)(2,4)$ $-1$ $72$ $9$ $(1,6,4,3,7,5,8,9,2)$ $0$ $72$ $9$ $(1,6,5,8,9,4,3,7,2)$ $0$ $54$ $12$ $(1,2,3,4)(5,8)(6,7,9)$ $0$ $54$ $12$ $(1,2,3,4)(5,8)(6,9,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.