# Properties

 Label 11.276...576.12t218.a.a Dimension $11$ Group $\PGL(2,11)$ Conductor $2.760\times 10^{20}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $11$ Group: $\PGL(2,11)$ Conductor: $$276\!\cdots\!576$$$$\medspace = 2^{14} \cdot 3^{10} \cdot 11^{11}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 12.2.276027291040300056576.1 Galois orbit size: $1$ Smallest permutation container: $\PGL(2,11)$ Parity: odd Determinant: 1.11.2t1.a.a Projective image: $\PSL(2,11).C_2$ Projective stem field: Galois closure of 12.2.276027291040300056576.1

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - 2x^{11} - 99x^{8} + 132x^{7} - 132x^{6} + 3267x^{4} - 2178x^{3} + 4356x^{2} - 1656x - 35793$$ x^12 - 2*x^11 - 99*x^8 + 132*x^7 - 132*x^6 + 3267*x^4 - 2178*x^3 + 4356*x^2 - 1656*x - 35793 .

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $$x^{6} + x^{4} + 82x^{3} + 80x^{2} + 15x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$44 a^{5} + 80 a^{4} + 39 a^{3} + 49 a^{2} + 66 a + 9 + \left(58 a^{5} + 49 a^{4} + 46 a^{3} + 37 a^{2} + 19 a + 45\right)\cdot 89 + \left(22 a^{5} + 29 a^{4} + 21 a^{3} + 74 a^{2} + 74\right)\cdot 89^{2} + \left(4 a^{5} + a^{4} + 2 a^{3} + 69 a^{2} + 59 a + 84\right)\cdot 89^{3} + \left(44 a^{5} + 87 a^{4} + 15 a^{3} + 66 a^{2} + 44 a + 67\right)\cdot 89^{4} + \left(52 a^{5} + 75 a^{4} + 38 a^{3} + 66 a^{2} + 11 a + 74\right)\cdot 89^{5} + \left(19 a^{5} + 26 a^{4} + 65 a^{3} + 28 a^{2} + 58 a + 85\right)\cdot 89^{6} + \left(52 a^{5} + 38 a^{4} + 3 a^{3} + 29 a^{2} + 54 a + 80\right)\cdot 89^{7} + \left(50 a^{5} + 44 a^{4} + 3 a^{3} + 78 a^{2} + 47 a + 11\right)\cdot 89^{8} + \left(46 a^{5} + 58 a^{4} + 49 a^{3} + 8 a^{2} + 11 a + 73\right)\cdot 89^{9} +O(89^{10})$$ 44*a^5 + 80*a^4 + 39*a^3 + 49*a^2 + 66*a + 9 + (58*a^5 + 49*a^4 + 46*a^3 + 37*a^2 + 19*a + 45)*89 + (22*a^5 + 29*a^4 + 21*a^3 + 74*a^2 + 74)*89^2 + (4*a^5 + a^4 + 2*a^3 + 69*a^2 + 59*a + 84)*89^3 + (44*a^5 + 87*a^4 + 15*a^3 + 66*a^2 + 44*a + 67)*89^4 + (52*a^5 + 75*a^4 + 38*a^3 + 66*a^2 + 11*a + 74)*89^5 + (19*a^5 + 26*a^4 + 65*a^3 + 28*a^2 + 58*a + 85)*89^6 + (52*a^5 + 38*a^4 + 3*a^3 + 29*a^2 + 54*a + 80)*89^7 + (50*a^5 + 44*a^4 + 3*a^3 + 78*a^2 + 47*a + 11)*89^8 + (46*a^5 + 58*a^4 + 49*a^3 + 8*a^2 + 11*a + 73)*89^9+O(89^10) $r_{ 2 }$ $=$ $$68 a^{5} + 2 a^{4} + 13 a^{3} + 48 a^{2} + 45 a + 25 + \left(52 a^{5} + 52 a^{4} + 52 a^{3} + 88 a^{2} + 24 a + 46\right)\cdot 89 + \left(5 a^{5} + 36 a^{4} + 19 a^{3} + 65 a^{2} + 32 a + 56\right)\cdot 89^{2} + \left(17 a^{5} + 15 a^{4} + 85 a^{3} + 18 a^{2} + 38 a + 59\right)\cdot 89^{3} + \left(20 a^{5} + 18 a^{4} + 71 a^{3} + 46 a^{2} + 43 a + 78\right)\cdot 89^{4} + \left(32 a^{5} + 44 a^{4} + 79 a^{3} + 34 a^{2} + 14 a + 61\right)\cdot 89^{5} + \left(13 a^{5} + 18 a^{4} + 31 a^{3} + 21 a^{2} + 52 a + 7\right)\cdot 89^{6} + \left(74 a^{5} + 54 a^{4} + 41 a^{3} + 5 a^{2} + 46 a + 77\right)\cdot 89^{7} + \left(32 a^{5} + 60 a^{4} + 46 a^{3} + 13 a^{2} + 62 a + 88\right)\cdot 89^{8} + \left(43 a^{5} + 39 a^{4} + 41 a^{3} + 69 a^{2} + 7 a + 78\right)\cdot 89^{9} +O(89^{10})$$ 68*a^5 + 2*a^4 + 13*a^3 + 48*a^2 + 45*a + 25 + (52*a^5 + 52*a^4 + 52*a^3 + 88*a^2 + 24*a + 46)*89 + (5*a^5 + 36*a^4 + 19*a^3 + 65*a^2 + 32*a + 56)*89^2 + (17*a^5 + 15*a^4 + 85*a^3 + 18*a^2 + 38*a + 59)*89^3 + (20*a^5 + 18*a^4 + 71*a^3 + 46*a^2 + 43*a + 78)*89^4 + (32*a^5 + 44*a^4 + 79*a^3 + 34*a^2 + 14*a + 61)*89^5 + (13*a^5 + 18*a^4 + 31*a^3 + 21*a^2 + 52*a + 7)*89^6 + (74*a^5 + 54*a^4 + 41*a^3 + 5*a^2 + 46*a + 77)*89^7 + (32*a^5 + 60*a^4 + 46*a^3 + 13*a^2 + 62*a + 88)*89^8 + (43*a^5 + 39*a^4 + 41*a^3 + 69*a^2 + 7*a + 78)*89^9+O(89^10) $r_{ 3 }$ $=$ $$16 a^{5} + 46 a^{4} + 26 a^{3} + 53 a^{2} + 44 a + 10 + \left(38 a^{5} + 12 a^{4} + 35 a^{3} + 35 a + 34\right)\cdot 89 + \left(71 a^{5} + 59 a^{4} + 26 a^{3} + 36 a^{2} + 65 a + 70\right)\cdot 89^{2} + \left(62 a^{5} + 6 a^{4} + 82 a^{3} + 20 a^{2} + 10 a + 52\right)\cdot 89^{3} + \left(76 a^{5} + 45 a^{4} + 10 a^{3} + 41 a^{2} + 20 a + 72\right)\cdot 89^{4} + \left(7 a^{5} + 15 a^{4} + 74 a^{3} + 48 a^{2} + 14 a + 28\right)\cdot 89^{5} + \left(63 a^{5} + 37 a^{4} + 8 a^{3} + 83 a^{2} + 74 a + 55\right)\cdot 89^{6} + \left(33 a^{5} + 52 a^{4} + 6 a^{3} + 34 a^{2} + 81 a + 77\right)\cdot 89^{7} + \left(78 a^{5} + 31 a^{4} + 86 a^{3} + 7 a^{2} + 27 a + 17\right)\cdot 89^{8} + \left(11 a^{5} + 3 a^{4} + 24 a^{3} + 82 a^{2} + 39 a + 53\right)\cdot 89^{9} +O(89^{10})$$ 16*a^5 + 46*a^4 + 26*a^3 + 53*a^2 + 44*a + 10 + (38*a^5 + 12*a^4 + 35*a^3 + 35*a + 34)*89 + (71*a^5 + 59*a^4 + 26*a^3 + 36*a^2 + 65*a + 70)*89^2 + (62*a^5 + 6*a^4 + 82*a^3 + 20*a^2 + 10*a + 52)*89^3 + (76*a^5 + 45*a^4 + 10*a^3 + 41*a^2 + 20*a + 72)*89^4 + (7*a^5 + 15*a^4 + 74*a^3 + 48*a^2 + 14*a + 28)*89^5 + (63*a^5 + 37*a^4 + 8*a^3 + 83*a^2 + 74*a + 55)*89^6 + (33*a^5 + 52*a^4 + 6*a^3 + 34*a^2 + 81*a + 77)*89^7 + (78*a^5 + 31*a^4 + 86*a^3 + 7*a^2 + 27*a + 17)*89^8 + (11*a^5 + 3*a^4 + 24*a^3 + 82*a^2 + 39*a + 53)*89^9+O(89^10) $r_{ 4 }$ $=$ $$15 a^{5} + 81 a^{4} + 76 a^{3} + 38 a^{2} + 48 a + 21 + \left(62 a^{5} + 47 a^{4} + 41 a^{3} + 29 a^{2} + 71 a + 71\right)\cdot 89 + \left(29 a^{5} + 39 a^{4} + 69 a^{3} + 68 a^{2} + 14 a + 81\right)\cdot 89^{2} + \left(26 a^{5} + 59 a^{4} + 37 a^{3} + 25 a^{2} + 54 a + 15\right)\cdot 89^{3} + \left(83 a^{5} + 79 a^{4} + 9 a^{3} + 32 a^{2} + 25 a + 66\right)\cdot 89^{4} + \left(77 a^{5} + 52 a^{4} + 31 a^{3} + 43 a^{2} + 72 a + 65\right)\cdot 89^{5} + \left(19 a^{5} + 24 a^{4} + 35 a^{2} + 33 a + 40\right)\cdot 89^{6} + \left(27 a^{5} + 68 a^{4} + 8 a^{3} + 48 a^{2} + 33 a + 22\right)\cdot 89^{7} + \left(6 a^{5} + 43 a^{4} + 17 a^{3} + 67 a^{2} + 67 a + 31\right)\cdot 89^{8} + \left(12 a^{5} + 28 a^{4} + 51 a^{3} + 82 a^{2} + 4 a + 48\right)\cdot 89^{9} +O(89^{10})$$ 15*a^5 + 81*a^4 + 76*a^3 + 38*a^2 + 48*a + 21 + (62*a^5 + 47*a^4 + 41*a^3 + 29*a^2 + 71*a + 71)*89 + (29*a^5 + 39*a^4 + 69*a^3 + 68*a^2 + 14*a + 81)*89^2 + (26*a^5 + 59*a^4 + 37*a^3 + 25*a^2 + 54*a + 15)*89^3 + (83*a^5 + 79*a^4 + 9*a^3 + 32*a^2 + 25*a + 66)*89^4 + (77*a^5 + 52*a^4 + 31*a^3 + 43*a^2 + 72*a + 65)*89^5 + (19*a^5 + 24*a^4 + 35*a^2 + 33*a + 40)*89^6 + (27*a^5 + 68*a^4 + 8*a^3 + 48*a^2 + 33*a + 22)*89^7 + (6*a^5 + 43*a^4 + 17*a^3 + 67*a^2 + 67*a + 31)*89^8 + (12*a^5 + 28*a^4 + 51*a^3 + 82*a^2 + 4*a + 48)*89^9+O(89^10) $r_{ 5 }$ $=$ $$18 a^{5} + 55 a^{4} + 42 a^{3} + 34 a^{2} + 45 a + 32 + \left(55 a^{5} + 24 a^{4} + 8 a^{3} + 2 a + 72\right)\cdot 89 + \left(41 a^{5} + 42 a^{4} + 10 a^{3} + 34 a^{2} + 86 a + 22\right)\cdot 89^{2} + \left(33 a^{5} + 9 a^{4} + 15 a^{3} + 64 a^{2} + 27 a + 13\right)\cdot 89^{3} + \left(39 a^{5} + 25 a^{4} + 69 a^{3} + 45 a^{2} + 2 a + 78\right)\cdot 89^{4} + \left(55 a^{5} + 26 a^{4} + 48 a^{3} + 58 a^{2} + 26 a + 2\right)\cdot 89^{5} + \left(63 a^{5} + 70 a^{4} + 40 a^{3} + 19 a^{2} + 76\right)\cdot 89^{6} + \left(36 a^{5} + 9 a^{4} + 64 a^{3} + 82 a^{2} + 31 a + 23\right)\cdot 89^{7} + \left(8 a^{5} + 22 a^{4} + 26 a^{3} + 48 a^{2} + 12 a + 74\right)\cdot 89^{8} + \left(33 a^{5} + 70 a^{4} + 84 a^{3} + 53 a^{2} + 83 a + 76\right)\cdot 89^{9} +O(89^{10})$$ 18*a^5 + 55*a^4 + 42*a^3 + 34*a^2 + 45*a + 32 + (55*a^5 + 24*a^4 + 8*a^3 + 2*a + 72)*89 + (41*a^5 + 42*a^4 + 10*a^3 + 34*a^2 + 86*a + 22)*89^2 + (33*a^5 + 9*a^4 + 15*a^3 + 64*a^2 + 27*a + 13)*89^3 + (39*a^5 + 25*a^4 + 69*a^3 + 45*a^2 + 2*a + 78)*89^4 + (55*a^5 + 26*a^4 + 48*a^3 + 58*a^2 + 26*a + 2)*89^5 + (63*a^5 + 70*a^4 + 40*a^3 + 19*a^2 + 76)*89^6 + (36*a^5 + 9*a^4 + 64*a^3 + 82*a^2 + 31*a + 23)*89^7 + (8*a^5 + 22*a^4 + 26*a^3 + 48*a^2 + 12*a + 74)*89^8 + (33*a^5 + 70*a^4 + 84*a^3 + 53*a^2 + 83*a + 76)*89^9+O(89^10) $r_{ 6 }$ $=$ $$51 a^{5} + 62 a^{4} + 28 a^{3} + 32 a^{2} + 5 a + 18 + \left(29 a^{5} + 34 a^{4} + a^{3} + 46 a^{2} + 49 a + 83\right)\cdot 89 + \left(18 a^{5} + 2 a^{4} + 88 a^{3} + 77 a^{2} + 51 a + 44\right)\cdot 89^{2} + \left(34 a^{5} + 47 a^{4} + 87 a^{3} + 76 a^{2} + 77 a + 44\right)\cdot 89^{3} + \left(29 a^{5} + 61 a^{4} + 37 a^{3} + 62 a^{2} + 82 a + 62\right)\cdot 89^{4} + \left(44 a^{5} + 51 a^{4} + 31 a^{3} + 9 a^{2} + 72 a + 55\right)\cdot 89^{5} + \left(79 a^{5} + 51 a^{4} + 18 a^{3} + 16 a^{2} + 4 a + 45\right)\cdot 89^{6} + \left(5 a^{5} + 45 a^{4} + 39 a^{3} + 13 a^{2} + 46 a + 76\right)\cdot 89^{7} + \left(65 a^{5} + 84 a^{4} + 28 a^{3} + 18 a^{2} + 86 a + 70\right)\cdot 89^{8} + \left(41 a^{5} + 70 a^{4} + 53 a^{3} + 79 a^{2} + 45 a + 1\right)\cdot 89^{9} +O(89^{10})$$ 51*a^5 + 62*a^4 + 28*a^3 + 32*a^2 + 5*a + 18 + (29*a^5 + 34*a^4 + a^3 + 46*a^2 + 49*a + 83)*89 + (18*a^5 + 2*a^4 + 88*a^3 + 77*a^2 + 51*a + 44)*89^2 + (34*a^5 + 47*a^4 + 87*a^3 + 76*a^2 + 77*a + 44)*89^3 + (29*a^5 + 61*a^4 + 37*a^3 + 62*a^2 + 82*a + 62)*89^4 + (44*a^5 + 51*a^4 + 31*a^3 + 9*a^2 + 72*a + 55)*89^5 + (79*a^5 + 51*a^4 + 18*a^3 + 16*a^2 + 4*a + 45)*89^6 + (5*a^5 + 45*a^4 + 39*a^3 + 13*a^2 + 46*a + 76)*89^7 + (65*a^5 + 84*a^4 + 28*a^3 + 18*a^2 + 86*a + 70)*89^8 + (41*a^5 + 70*a^4 + 53*a^3 + 79*a^2 + 45*a + 1)*89^9+O(89^10) $r_{ 7 }$ $=$ $$5 a^{5} + 20 a^{4} + 77 a^{3} + 2 a^{2} + 65 a + \left(29 a^{5} + 42 a^{4} + 81 a^{3} + 71 a^{2} + 16 a + 46\right)\cdot 89 + \left(54 a^{5} + 19 a^{4} + 85 a^{3} + 18 a^{2} + 59 a + 37\right)\cdot 89^{2} + \left(76 a^{5} + 55 a^{4} + 39 a^{3} + 8 a^{2} + 67 a + 41\right)\cdot 89^{3} + \left(52 a^{5} + 13 a^{4} + 80 a^{3} + 83 a^{2} + 51 a + 5\right)\cdot 89^{4} + \left(49 a^{5} + 52 a^{4} + 21 a^{3} + 8 a^{2} + 20 a + 53\right)\cdot 89^{5} + \left(87 a^{5} + 16 a^{4} + 60 a^{3} + 57 a^{2} + 73 a + 8\right)\cdot 89^{6} + \left(34 a^{5} + 46 a^{4} + 88 a^{3} + 71 a^{2} + 41 a + 48\right)\cdot 89^{7} + \left(77 a^{5} + 8 a^{4} + 70 a^{3} + 43 a^{2} + 38 a + 9\right)\cdot 89^{8} + \left(76 a^{5} + 72 a^{4} + 51 a^{3} + 10 a^{2} + 30 a + 76\right)\cdot 89^{9} +O(89^{10})$$ 5*a^5 + 20*a^4 + 77*a^3 + 2*a^2 + 65*a + (29*a^5 + 42*a^4 + 81*a^3 + 71*a^2 + 16*a + 46)*89 + (54*a^5 + 19*a^4 + 85*a^3 + 18*a^2 + 59*a + 37)*89^2 + (76*a^5 + 55*a^4 + 39*a^3 + 8*a^2 + 67*a + 41)*89^3 + (52*a^5 + 13*a^4 + 80*a^3 + 83*a^2 + 51*a + 5)*89^4 + (49*a^5 + 52*a^4 + 21*a^3 + 8*a^2 + 20*a + 53)*89^5 + (87*a^5 + 16*a^4 + 60*a^3 + 57*a^2 + 73*a + 8)*89^6 + (34*a^5 + 46*a^4 + 88*a^3 + 71*a^2 + 41*a + 48)*89^7 + (77*a^5 + 8*a^4 + 70*a^3 + 43*a^2 + 38*a + 9)*89^8 + (76*a^5 + 72*a^4 + 51*a^3 + 10*a^2 + 30*a + 76)*89^9+O(89^10) $r_{ 8 }$ $=$ $$60 a^{5} + 42 a^{4} + 26 a^{2} + 73 a + 34 + \left(75 a^{5} + 84 a^{4} + 34 a^{3} + 27 a^{2} + 78 a + 4\right)\cdot 89 + \left(49 a^{5} + 10 a^{4} + 68 a^{3} + 53 a^{2} + 2 a + 18\right)\cdot 89^{2} + \left(46 a^{5} + 47 a^{4} + 67 a^{3} + 63 a^{2} + 46 a + 75\right)\cdot 89^{3} + \left(40 a^{5} + 77 a^{4} + 29 a^{3} + 43 a^{2} + 45 a + 61\right)\cdot 89^{4} + \left(26 a^{5} + 59 a^{4} + 61 a^{3} + 13 a^{2} + 31 a + 34\right)\cdot 89^{5} + \left(77 a^{5} + 11 a^{4} + 72 a^{3} + 70 a^{2} + 54 a + 49\right)\cdot 89^{6} + \left(30 a^{5} + 53 a^{4} + 25 a^{3} + 68 a^{2} + 67 a + 40\right)\cdot 89^{7} + \left(48 a^{5} + 74 a^{4} + 17 a^{3} + 18 a^{2} + 79 a + 45\right)\cdot 89^{8} + \left(77 a^{5} + 66 a^{4} + 72 a^{3} + 51 a^{2} + 2 a + 1\right)\cdot 89^{9} +O(89^{10})$$ 60*a^5 + 42*a^4 + 26*a^2 + 73*a + 34 + (75*a^5 + 84*a^4 + 34*a^3 + 27*a^2 + 78*a + 4)*89 + (49*a^5 + 10*a^4 + 68*a^3 + 53*a^2 + 2*a + 18)*89^2 + (46*a^5 + 47*a^4 + 67*a^3 + 63*a^2 + 46*a + 75)*89^3 + (40*a^5 + 77*a^4 + 29*a^3 + 43*a^2 + 45*a + 61)*89^4 + (26*a^5 + 59*a^4 + 61*a^3 + 13*a^2 + 31*a + 34)*89^5 + (77*a^5 + 11*a^4 + 72*a^3 + 70*a^2 + 54*a + 49)*89^6 + (30*a^5 + 53*a^4 + 25*a^3 + 68*a^2 + 67*a + 40)*89^7 + (48*a^5 + 74*a^4 + 17*a^3 + 18*a^2 + 79*a + 45)*89^8 + (77*a^5 + 66*a^4 + 72*a^3 + 51*a^2 + 2*a + 1)*89^9+O(89^10) $r_{ 9 }$ $=$ $$74 a^{5} + 77 a^{4} + 23 a^{3} + 63 a + 54 + \left(12 a^{5} + 25 a^{4} + 17 a^{3} + 42 a^{2} + 2 a + 66\right)\cdot 89 + \left(63 a^{5} + 22 a^{4} + 45 a^{3} + 18 a^{2} + 18 a + 56\right)\cdot 89^{2} + \left(59 a^{5} + 52 a^{4} + 78 a^{3} + 86 a^{2} + 45 a + 60\right)\cdot 89^{3} + \left(32 a^{5} + 25 a^{4} + 58 a^{3} + 74 a^{2} + 61 a + 19\right)\cdot 89^{4} + \left(9 a^{5} + 19 a^{4} + 80 a^{3} + 5 a^{2} + 85 a + 28\right)\cdot 89^{5} + \left(6 a^{5} + 67 a^{4} + 27 a^{3} + 6 a^{2} + 43 a + 67\right)\cdot 89^{6} + \left(41 a^{5} + 22 a^{4} + 12 a^{3} + 57 a^{2} + 63 a + 47\right)\cdot 89^{7} + \left(68 a^{5} + 47 a^{4} + 43 a^{3} + 16 a^{2} + 10 a + 30\right)\cdot 89^{8} + \left(10 a^{5} + 26 a^{4} + 27 a^{3} + 45 a^{2} + 86 a + 25\right)\cdot 89^{9} +O(89^{10})$$ 74*a^5 + 77*a^4 + 23*a^3 + 63*a + 54 + (12*a^5 + 25*a^4 + 17*a^3 + 42*a^2 + 2*a + 66)*89 + (63*a^5 + 22*a^4 + 45*a^3 + 18*a^2 + 18*a + 56)*89^2 + (59*a^5 + 52*a^4 + 78*a^3 + 86*a^2 + 45*a + 60)*89^3 + (32*a^5 + 25*a^4 + 58*a^3 + 74*a^2 + 61*a + 19)*89^4 + (9*a^5 + 19*a^4 + 80*a^3 + 5*a^2 + 85*a + 28)*89^5 + (6*a^5 + 67*a^4 + 27*a^3 + 6*a^2 + 43*a + 67)*89^6 + (41*a^5 + 22*a^4 + 12*a^3 + 57*a^2 + 63*a + 47)*89^7 + (68*a^5 + 47*a^4 + 43*a^3 + 16*a^2 + 10*a + 30)*89^8 + (10*a^5 + 26*a^4 + 27*a^3 + 45*a^2 + 86*a + 25)*89^9+O(89^10) $r_{ 10 }$ $=$ $$88 a^{5} + 83 a^{4} + 62 a^{3} + 15 a^{2} + 8 a + 53 + \left(5 a^{5} + 8 a^{4} + 76 a^{3} + 48 a^{2} + 34 a + 25\right)\cdot 89 + \left(73 a^{5} + 39 a^{4} + 55 a^{3} + 84 a^{2} + 31 a + 51\right)\cdot 89^{2} + \left(50 a^{5} + 27 a^{4} + 48 a^{3} + 82 a^{2} + 39 a + 56\right)\cdot 89^{3} + \left(45 a^{5} + 86 a^{4} + 48 a^{3} + 54 a^{2} + 38 a + 71\right)\cdot 89^{4} + \left(39 a^{5} + 74 a^{4} + 25 a^{3} + 29 a^{2} + 63 a + 3\right)\cdot 89^{5} + \left(16 a^{5} + 61 a^{4} + 79 a^{3} + 76 a^{2} + 9 a + 72\right)\cdot 89^{6} + \left(46 a^{5} + 25 a^{4} + 69 a^{3} + 65 a^{2} + 69 a + 80\right)\cdot 89^{7} + \left(71 a^{5} + 3 a^{4} + 13 a^{3} + 80 a^{2} + 42 a + 61\right)\cdot 89^{8} + \left(84 a^{5} + 60 a^{4} + 27 a^{3} + 51 a^{2} + 42 a + 30\right)\cdot 89^{9} +O(89^{10})$$ 88*a^5 + 83*a^4 + 62*a^3 + 15*a^2 + 8*a + 53 + (5*a^5 + 8*a^4 + 76*a^3 + 48*a^2 + 34*a + 25)*89 + (73*a^5 + 39*a^4 + 55*a^3 + 84*a^2 + 31*a + 51)*89^2 + (50*a^5 + 27*a^4 + 48*a^3 + 82*a^2 + 39*a + 56)*89^3 + (45*a^5 + 86*a^4 + 48*a^3 + 54*a^2 + 38*a + 71)*89^4 + (39*a^5 + 74*a^4 + 25*a^3 + 29*a^2 + 63*a + 3)*89^5 + (16*a^5 + 61*a^4 + 79*a^3 + 76*a^2 + 9*a + 72)*89^6 + (46*a^5 + 25*a^4 + 69*a^3 + 65*a^2 + 69*a + 80)*89^7 + (71*a^5 + 3*a^4 + 13*a^3 + 80*a^2 + 42*a + 61)*89^8 + (84*a^5 + 60*a^4 + 27*a^3 + 51*a^2 + 42*a + 30)*89^9+O(89^10) $r_{ 11 }$ $=$ $$16 a^{5} + 28 a^{4} + 60 a^{3} + 40 a^{2} + 31 a + 46 + \left(88 a^{5} + 12 a^{4} + 17 a^{3} + 43 a^{2} + 31 a + 32\right)\cdot 89 + \left(12 a^{5} + 84 a^{4} + 79 a^{3} + 50 a^{2} + 53 a + 33\right)\cdot 89^{2} + \left(70 a^{5} + 12 a^{4} + 62 a^{3} + 24 a^{2} + 29 a + 50\right)\cdot 89^{3} + \left(43 a^{5} + 49 a^{4} + 8 a^{3} + 3 a^{2} + 73 a + 51\right)\cdot 89^{4} + \left(29 a^{5} + 27 a^{4} + 42 a^{3} + 12 a^{2} + 18 a + 87\right)\cdot 89^{5} + \left(18 a^{5} + 80 a^{4} + 16 a^{3} + 46 a^{2} + 61 a + 50\right)\cdot 89^{6} + \left(15 a^{5} + 36 a^{4} + 53 a^{3} + 25 a^{2} + 24 a + 59\right)\cdot 89^{7} + \left(82 a^{5} + 52 a^{4} + 60 a^{3} + 64 a^{2} + 69 a + 68\right)\cdot 89^{8} + \left(86 a^{5} + 63 a^{4} + 47 a^{3} + 24 a^{2} + 80 a + 72\right)\cdot 89^{9} +O(89^{10})$$ 16*a^5 + 28*a^4 + 60*a^3 + 40*a^2 + 31*a + 46 + (88*a^5 + 12*a^4 + 17*a^3 + 43*a^2 + 31*a + 32)*89 + (12*a^5 + 84*a^4 + 79*a^3 + 50*a^2 + 53*a + 33)*89^2 + (70*a^5 + 12*a^4 + 62*a^3 + 24*a^2 + 29*a + 50)*89^3 + (43*a^5 + 49*a^4 + 8*a^3 + 3*a^2 + 73*a + 51)*89^4 + (29*a^5 + 27*a^4 + 42*a^3 + 12*a^2 + 18*a + 87)*89^5 + (18*a^5 + 80*a^4 + 16*a^3 + 46*a^2 + 61*a + 50)*89^6 + (15*a^5 + 36*a^4 + 53*a^3 + 25*a^2 + 24*a + 59)*89^7 + (82*a^5 + 52*a^4 + 60*a^3 + 64*a^2 + 69*a + 68)*89^8 + (86*a^5 + 63*a^4 + 47*a^3 + 24*a^2 + 80*a + 72)*89^9+O(89^10) $r_{ 12 }$ $=$ $$79 a^{5} + 47 a^{4} + 88 a^{3} + 19 a^{2} + 41 a + 56 + \left(25 a^{5} + 49 a^{4} + 31 a^{3} + 10 a^{2} + 78 a + 6\right)\cdot 89 + \left(2 a^{5} + 59 a^{4} + 53 a^{3} + 41 a^{2} + 29 a + 75\right)\cdot 89^{2} + \left(52 a^{5} + 21 a^{4} + 14 a^{3} + 81 a^{2} + 38 a + 67\right)\cdot 89^{3} + \left(25 a^{5} + 54 a^{4} + 4 a^{3} + 67 a^{2} + 44 a + 75\right)\cdot 89^{4} + \left(20 a^{5} + 33 a^{4} + 88 a^{3} + 24 a^{2} + 13 a + 36\right)\cdot 89^{5} + \left(69 a^{5} + 67 a^{4} + 22 a^{3} + 73 a^{2} + 68 a + 63\right)\cdot 89^{6} + \left(46 a^{5} + 80 a^{4} + 32 a^{3} + 31 a^{2} + 62 a + 76\right)\cdot 89^{7} + \left(33 a^{5} + 60 a^{4} + 31 a^{3} + 76 a^{2} + 77 a + 22\right)\cdot 89^{8} + \left(8 a^{5} + 62 a^{4} + 3 a^{3} + 63 a^{2} + 9 a + 84\right)\cdot 89^{9} +O(89^{10})$$ 79*a^5 + 47*a^4 + 88*a^3 + 19*a^2 + 41*a + 56 + (25*a^5 + 49*a^4 + 31*a^3 + 10*a^2 + 78*a + 6)*89 + (2*a^5 + 59*a^4 + 53*a^3 + 41*a^2 + 29*a + 75)*89^2 + (52*a^5 + 21*a^4 + 14*a^3 + 81*a^2 + 38*a + 67)*89^3 + (25*a^5 + 54*a^4 + 4*a^3 + 67*a^2 + 44*a + 75)*89^4 + (20*a^5 + 33*a^4 + 88*a^3 + 24*a^2 + 13*a + 36)*89^5 + (69*a^5 + 67*a^4 + 22*a^3 + 73*a^2 + 68*a + 63)*89^6 + (46*a^5 + 80*a^4 + 32*a^3 + 31*a^2 + 62*a + 76)*89^7 + (33*a^5 + 60*a^4 + 31*a^3 + 76*a^2 + 77*a + 22)*89^8 + (8*a^5 + 62*a^4 + 3*a^3 + 63*a^2 + 9*a + 84)*89^9+O(89^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

 Cycle notation $(2,8,6,12,3,11,7,4,9,5)$ $(1,10)(2,4)(3,12)(5,9)(6,11)(7,8)$ $(1,8,6,5,12,11,2,9,3,4,7)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 12 }$ Character value $1$ $1$ $()$ $11$ $55$ $2$ $(1,10)(2,4)(3,12)(5,9)(6,11)(7,8)$ $-1$ $66$ $2$ $(2,11)(3,5)(4,6)(7,8)(9,12)$ $1$ $110$ $3$ $(1,7,10)(2,5,3)(4,8,11)(6,9,12)$ $-1$ $110$ $4$ $(1,11,7,10)(2,9,6,4)(3,5,12,8)$ $-1$ $132$ $5$ $(2,6,3,7,9)(4,5,8,12,11)$ $1$ $132$ $5$ $(1,3,7,9,10)(2,8,6,4,11)$ $1$ $110$ $6$ $(1,8,3,7,2,9)(4,5,6,10,11,12)$ $-1$ $132$ $10$ $(2,8,6,12,3,11,7,4,9,5)$ $1$ $132$ $10$ $(2,12,7,5,6,11,9,8,3,4)$ $1$ $120$ $11$ $(1,8,6,5,12,11,2,9,3,4,7)$ $0$ $110$ $12$ $(1,11,8,12,3,4,7,5,2,6,9,10)$ $-1$ $110$ $12$ $(1,4,9,12,2,11,7,10,3,6,8,5)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.