# Properties

 Label 11.137...739.12t218.a.a Dimension $11$ Group $\PGL(2,11)$ Conductor $1.372\times 10^{27}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $11$ Group: $\PGL(2,11)$ Conductor: $$137\!\cdots\!739$$$$\medspace = 11^{11} \cdot 37^{10}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 12.2.1371945240568483487545135739.1 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 12.2.1371945240568483487545135739.1

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - 22 x^{10} - 55 x^{9} - 165 x^{8} - 132 x^{7} + 55 x^{6} - 66 x^{5} - 495 x^{4} - 385 x^{3} - 319 x^{2} - 55 x - 1331$$ x^12 - 22*x^10 - 55*x^9 - 165*x^8 - 132*x^7 + 55*x^6 - 66*x^5 - 495*x^4 - 385*x^3 - 319*x^2 - 55*x - 1331 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{6} + 19x^{3} + 16x^{2} + 8x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$27 a^{5} + 9 a^{4} + 18 a^{3} + 5 a^{2} + 15 a + 19 + \left(3 a^{5} + 2 a^{4} + 27 a^{3} + 27 a^{2} + 5 a + 24\right)\cdot 31 + \left(17 a^{5} + 8 a^{4} + 7 a^{3} + 22 a^{2} + 30 a + 3\right)\cdot 31^{2} + \left(26 a^{5} + 2 a^{4} + 30 a^{3} + 3 a^{2} + 22 a + 15\right)\cdot 31^{3} + \left(6 a^{5} + a^{4} + 13 a^{3} + 9 a^{2} + 25 a + 19\right)\cdot 31^{4} + \left(25 a^{5} + 8 a^{4} + 3 a^{3} + 19 a^{2} + 7 a + 9\right)\cdot 31^{5} + \left(13 a^{5} + 19 a^{4} + 4 a^{2} + 21 a\right)\cdot 31^{6} + \left(8 a^{5} + 19 a^{4} + 17 a^{3} + 25 a^{2} + 18 a + 8\right)\cdot 31^{7} + \left(28 a^{5} + 2 a^{4} + 23 a^{3} + 19 a^{2} + 19 a + 23\right)\cdot 31^{8} + \left(21 a^{5} + 27 a^{3} + 19 a^{2} + 20 a + 24\right)\cdot 31^{9} +O(31^{10})$$ 27*a^5 + 9*a^4 + 18*a^3 + 5*a^2 + 15*a + 19 + (3*a^5 + 2*a^4 + 27*a^3 + 27*a^2 + 5*a + 24)*31 + (17*a^5 + 8*a^4 + 7*a^3 + 22*a^2 + 30*a + 3)*31^2 + (26*a^5 + 2*a^4 + 30*a^3 + 3*a^2 + 22*a + 15)*31^3 + (6*a^5 + a^4 + 13*a^3 + 9*a^2 + 25*a + 19)*31^4 + (25*a^5 + 8*a^4 + 3*a^3 + 19*a^2 + 7*a + 9)*31^5 + (13*a^5 + 19*a^4 + 4*a^2 + 21*a)*31^6 + (8*a^5 + 19*a^4 + 17*a^3 + 25*a^2 + 18*a + 8)*31^7 + (28*a^5 + 2*a^4 + 23*a^3 + 19*a^2 + 19*a + 23)*31^8 + (21*a^5 + 27*a^3 + 19*a^2 + 20*a + 24)*31^9+O(31^10) $r_{ 2 }$ $=$ $$23 a^{5} + 30 a^{4} + 27 a^{2} + 8 a + 4 + \left(24 a^{5} + 4 a^{4} + 21 a^{3} + 12 a^{2} + 17 a + 16\right)\cdot 31 + \left(21 a^{5} + 9 a^{4} + 13 a^{3} + 2 a^{2} + 16 a + 8\right)\cdot 31^{2} + \left(a^{5} + 14 a^{4} + 2 a^{3} + 19 a^{2} + 13 a + 13\right)\cdot 31^{3} + \left(19 a^{5} + 17 a^{4} + 2 a^{3} + 6 a^{2} + 9 a + 2\right)\cdot 31^{4} + \left(9 a^{5} + 6 a^{4} + 3 a^{3} + 19 a^{2} + 11 a + 5\right)\cdot 31^{5} + \left(11 a^{5} + 13 a^{4} + 29 a^{3} + 12 a^{2} + 24 a + 29\right)\cdot 31^{6} + \left(30 a^{5} + 13 a^{4} + 25 a^{3} + 7 a^{2} + 23 a + 17\right)\cdot 31^{7} + \left(8 a^{5} + 2 a^{4} + 3 a^{3} + 16 a^{2} + 25\right)\cdot 31^{8} + \left(20 a^{5} + 4 a^{4} + 14 a^{3} + 7 a^{2} + 24 a + 14\right)\cdot 31^{9} +O(31^{10})$$ 23*a^5 + 30*a^4 + 27*a^2 + 8*a + 4 + (24*a^5 + 4*a^4 + 21*a^3 + 12*a^2 + 17*a + 16)*31 + (21*a^5 + 9*a^4 + 13*a^3 + 2*a^2 + 16*a + 8)*31^2 + (a^5 + 14*a^4 + 2*a^3 + 19*a^2 + 13*a + 13)*31^3 + (19*a^5 + 17*a^4 + 2*a^3 + 6*a^2 + 9*a + 2)*31^4 + (9*a^5 + 6*a^4 + 3*a^3 + 19*a^2 + 11*a + 5)*31^5 + (11*a^5 + 13*a^4 + 29*a^3 + 12*a^2 + 24*a + 29)*31^6 + (30*a^5 + 13*a^4 + 25*a^3 + 7*a^2 + 23*a + 17)*31^7 + (8*a^5 + 2*a^4 + 3*a^3 + 16*a^2 + 25)*31^8 + (20*a^5 + 4*a^4 + 14*a^3 + 7*a^2 + 24*a + 14)*31^9+O(31^10) $r_{ 3 }$ $=$ $$2 a^{5} + 22 a^{4} + 16 a^{3} + 13 a^{2} + 7 a + 3 + \left(25 a^{4} + 12 a^{2} + 14 a + 28\right)\cdot 31 + \left(26 a^{5} + 9 a^{4} + 4 a^{3} + 22 a^{2} + 28 a + 13\right)\cdot 31^{2} + \left(20 a^{5} + 25 a^{4} + 5 a^{3} + 30 a^{2} + 3 a + 20\right)\cdot 31^{3} + \left(4 a^{5} + 27 a^{4} + 19 a^{3} + 29 a^{2} + 11 a + 3\right)\cdot 31^{4} + \left(16 a^{5} + 14 a^{4} + 17 a^{3} + 26 a^{2} + 19 a + 7\right)\cdot 31^{5} + \left(22 a^{5} + 26 a^{4} + 4 a^{3} + 14 a^{2} + 5 a + 7\right)\cdot 31^{6} + \left(11 a^{5} + 12 a^{4} + 23 a^{3} + 5 a^{2} + 21\right)\cdot 31^{7} + \left(3 a^{5} + 19 a^{4} + 11 a^{3} + 27 a^{2} + 19 a + 1\right)\cdot 31^{8} + \left(14 a^{5} + 3 a^{4} + 21 a^{3} + 12 a^{2} + 20 a + 17\right)\cdot 31^{9} +O(31^{10})$$ 2*a^5 + 22*a^4 + 16*a^3 + 13*a^2 + 7*a + 3 + (25*a^4 + 12*a^2 + 14*a + 28)*31 + (26*a^5 + 9*a^4 + 4*a^3 + 22*a^2 + 28*a + 13)*31^2 + (20*a^5 + 25*a^4 + 5*a^3 + 30*a^2 + 3*a + 20)*31^3 + (4*a^5 + 27*a^4 + 19*a^3 + 29*a^2 + 11*a + 3)*31^4 + (16*a^5 + 14*a^4 + 17*a^3 + 26*a^2 + 19*a + 7)*31^5 + (22*a^5 + 26*a^4 + 4*a^3 + 14*a^2 + 5*a + 7)*31^6 + (11*a^5 + 12*a^4 + 23*a^3 + 5*a^2 + 21)*31^7 + (3*a^5 + 19*a^4 + 11*a^3 + 27*a^2 + 19*a + 1)*31^8 + (14*a^5 + 3*a^4 + 21*a^3 + 12*a^2 + 20*a + 17)*31^9+O(31^10) $r_{ 4 }$ $=$ $$27 a^{5} + 26 a^{4} + 9 a^{3} + 21 a + 15 + \left(8 a^{5} + 20 a^{4} + 13 a^{3} + 20 a^{2} + 22 a + 18\right)\cdot 31 + \left(29 a^{5} + 7 a^{4} + 2 a^{3} + 3 a^{2} + 28 a + 13\right)\cdot 31^{2} + \left(5 a^{5} + 18 a^{4} + 26 a^{3} + 17 a^{2} + 9 a + 25\right)\cdot 31^{3} + \left(23 a^{5} + 8 a^{4} + 6 a^{3} + 8 a^{2} + 30 a + 21\right)\cdot 31^{4} + \left(7 a^{5} + 10 a^{4} + 14 a^{3} + 12 a^{2} + 15 a + 25\right)\cdot 31^{5} + \left(27 a^{5} + 9 a^{4} + 23 a^{3} + 14 a^{2} + 30 a + 11\right)\cdot 31^{6} + \left(5 a^{5} + 23 a^{4} + 26 a^{3} + 26 a^{2} + 4 a + 10\right)\cdot 31^{7} + \left(7 a^{5} + 6 a^{4} + 14 a^{3} + 11 a^{2} + 11 a + 1\right)\cdot 31^{8} + \left(17 a^{5} + 8 a^{4} + 6 a^{3} + 20 a^{2} + 4 a + 30\right)\cdot 31^{9} +O(31^{10})$$ 27*a^5 + 26*a^4 + 9*a^3 + 21*a + 15 + (8*a^5 + 20*a^4 + 13*a^3 + 20*a^2 + 22*a + 18)*31 + (29*a^5 + 7*a^4 + 2*a^3 + 3*a^2 + 28*a + 13)*31^2 + (5*a^5 + 18*a^4 + 26*a^3 + 17*a^2 + 9*a + 25)*31^3 + (23*a^5 + 8*a^4 + 6*a^3 + 8*a^2 + 30*a + 21)*31^4 + (7*a^5 + 10*a^4 + 14*a^3 + 12*a^2 + 15*a + 25)*31^5 + (27*a^5 + 9*a^4 + 23*a^3 + 14*a^2 + 30*a + 11)*31^6 + (5*a^5 + 23*a^4 + 26*a^3 + 26*a^2 + 4*a + 10)*31^7 + (7*a^5 + 6*a^4 + 14*a^3 + 11*a^2 + 11*a + 1)*31^8 + (17*a^5 + 8*a^4 + 6*a^3 + 20*a^2 + 4*a + 30)*31^9+O(31^10) $r_{ 5 }$ $=$ $$4 a^{5} + 17 a^{3} + 25 a^{2} + 25 a + 22 + \left(21 a^{5} + 18 a^{4} + 4 a^{3} + 12 a^{2} + 15 a + 12\right)\cdot 31 + \left(24 a^{5} + 13 a^{4} + 11 a^{3} + 27 a + 25\right)\cdot 31^{2} + \left(11 a^{5} + a^{4} + 18 a^{3} + 19 a^{2} + 25 a + 28\right)\cdot 31^{3} + \left(18 a^{5} + 12 a^{4} + 14 a^{3} + 4 a^{2} + 22 a + 27\right)\cdot 31^{4} + \left(9 a^{5} + 7 a^{4} + 14 a^{3} + a^{2} + a + 19\right)\cdot 31^{5} + \left(29 a^{5} + 14 a^{4} + 15 a^{3} + 24 a^{2} + 3 a + 22\right)\cdot 31^{6} + \left(11 a^{5} + 24 a^{4} + 21 a^{3} + 18 a^{2} + 7 a + 28\right)\cdot 31^{7} + \left(16 a^{5} + 19 a^{4} + 9 a^{3} + 17 a^{2} + 5 a + 7\right)\cdot 31^{8} + \left(28 a^{5} + 9 a^{4} + 9 a^{3} + 4 a^{2} + 6 a + 18\right)\cdot 31^{9} +O(31^{10})$$ 4*a^5 + 17*a^3 + 25*a^2 + 25*a + 22 + (21*a^5 + 18*a^4 + 4*a^3 + 12*a^2 + 15*a + 12)*31 + (24*a^5 + 13*a^4 + 11*a^3 + 27*a + 25)*31^2 + (11*a^5 + a^4 + 18*a^3 + 19*a^2 + 25*a + 28)*31^3 + (18*a^5 + 12*a^4 + 14*a^3 + 4*a^2 + 22*a + 27)*31^4 + (9*a^5 + 7*a^4 + 14*a^3 + a^2 + a + 19)*31^5 + (29*a^5 + 14*a^4 + 15*a^3 + 24*a^2 + 3*a + 22)*31^6 + (11*a^5 + 24*a^4 + 21*a^3 + 18*a^2 + 7*a + 28)*31^7 + (16*a^5 + 19*a^4 + 9*a^3 + 17*a^2 + 5*a + 7)*31^8 + (28*a^5 + 9*a^4 + 9*a^3 + 4*a^2 + 6*a + 18)*31^9+O(31^10) $r_{ 6 }$ $=$ $$7 a^{5} + 8 a^{4} + 30 a^{3} + 13 a^{2} + 26 a + 8 + \left(a^{5} + 11 a^{4} + 6 a^{3} + 7 a^{2} + 23 a + 22\right)\cdot 31 + \left(16 a^{5} + 12 a^{4} + 23 a^{3} + 11 a^{2} + 25 a + 12\right)\cdot 31^{2} + \left(3 a^{5} + 17 a^{4} + 26 a^{3} + 29 a^{2} + 23 a + 1\right)\cdot 31^{3} + \left(8 a^{5} + 2 a^{4} + 15 a^{3} + 25 a^{2} + 24 a + 15\right)\cdot 31^{4} + \left(26 a^{5} + 5 a^{4} + 10 a^{3} + 24 a^{2} + 6 a + 27\right)\cdot 31^{5} + \left(25 a^{5} + 26 a^{4} + 14 a^{3} + 25 a^{2} + 21 a + 17\right)\cdot 31^{6} + \left(8 a^{5} + 7 a^{4} + 19 a^{3} + 9 a^{2} + 16 a + 2\right)\cdot 31^{7} + \left(20 a^{5} + 21 a^{4} + 30 a^{3} + 19 a^{2} + 29 a + 21\right)\cdot 31^{8} + \left(26 a^{5} + 2 a^{4} + 6 a^{3} + 21 a^{2} + 10 a + 7\right)\cdot 31^{9} +O(31^{10})$$ 7*a^5 + 8*a^4 + 30*a^3 + 13*a^2 + 26*a + 8 + (a^5 + 11*a^4 + 6*a^3 + 7*a^2 + 23*a + 22)*31 + (16*a^5 + 12*a^4 + 23*a^3 + 11*a^2 + 25*a + 12)*31^2 + (3*a^5 + 17*a^4 + 26*a^3 + 29*a^2 + 23*a + 1)*31^3 + (8*a^5 + 2*a^4 + 15*a^3 + 25*a^2 + 24*a + 15)*31^4 + (26*a^5 + 5*a^4 + 10*a^3 + 24*a^2 + 6*a + 27)*31^5 + (25*a^5 + 26*a^4 + 14*a^3 + 25*a^2 + 21*a + 17)*31^6 + (8*a^5 + 7*a^4 + 19*a^3 + 9*a^2 + 16*a + 2)*31^7 + (20*a^5 + 21*a^4 + 30*a^3 + 19*a^2 + 29*a + 21)*31^8 + (26*a^5 + 2*a^4 + 6*a^3 + 21*a^2 + 10*a + 7)*31^9+O(31^10) $r_{ 7 }$ $=$ $$4 a^{5} + 6 a^{4} + 29 a^{3} + 3 a^{2} + a + 26 + \left(18 a^{5} + 12 a^{4} + 16 a^{3} + 15 a^{2} + 5 a + 20\right)\cdot 31 + \left(14 a^{5} + 4 a^{4} + 2 a^{3} + 23 a^{2} + 4 a + 15\right)\cdot 31^{2} + \left(17 a^{5} + 4 a^{4} + 25 a^{3} + 6 a^{2} + 30 a + 18\right)\cdot 31^{3} + \left(23 a^{5} + 20 a^{4} + 26 a^{3} + 28 a^{2} + 8 a + 6\right)\cdot 31^{4} + \left(7 a^{5} + a^{4} + 25 a^{3} + 21 a^{2} + a + 12\right)\cdot 31^{5} + \left(20 a^{5} + 7 a^{4} + 29 a^{3} + 26 a^{2} + 26 a + 9\right)\cdot 31^{6} + \left(3 a^{5} + 10 a^{4} + 19 a^{3} + 6 a^{2} + 13 a + 7\right)\cdot 31^{7} + \left(5 a^{5} + a^{4} + 2 a^{3} + 30 a^{2} + 6 a + 23\right)\cdot 31^{8} + \left(25 a^{5} + 21 a^{4} + 26 a^{3} + 29 a^{2} + 6 a + 26\right)\cdot 31^{9} +O(31^{10})$$ 4*a^5 + 6*a^4 + 29*a^3 + 3*a^2 + a + 26 + (18*a^5 + 12*a^4 + 16*a^3 + 15*a^2 + 5*a + 20)*31 + (14*a^5 + 4*a^4 + 2*a^3 + 23*a^2 + 4*a + 15)*31^2 + (17*a^5 + 4*a^4 + 25*a^3 + 6*a^2 + 30*a + 18)*31^3 + (23*a^5 + 20*a^4 + 26*a^3 + 28*a^2 + 8*a + 6)*31^4 + (7*a^5 + a^4 + 25*a^3 + 21*a^2 + a + 12)*31^5 + (20*a^5 + 7*a^4 + 29*a^3 + 26*a^2 + 26*a + 9)*31^6 + (3*a^5 + 10*a^4 + 19*a^3 + 6*a^2 + 13*a + 7)*31^7 + (5*a^5 + a^4 + 2*a^3 + 30*a^2 + 6*a + 23)*31^8 + (25*a^5 + 21*a^4 + 26*a^3 + 29*a^2 + 6*a + 26)*31^9+O(31^10) $r_{ 8 }$ $=$ $$19 a^{5} + 24 a^{4} + 3 a^{3} + 7 a^{2} + 7 a + 9 + \left(9 a^{5} + 22 a^{4} + 19 a^{3} + 21 a^{2} + 28 a + 30\right)\cdot 31 + \left(22 a^{5} + 11 a^{4} + 26 a^{3} + 6 a + 27\right)\cdot 31^{2} + \left(27 a^{5} + 26 a^{4} + 22 a^{3} + 6 a^{2} + 15 a + 23\right)\cdot 31^{3} + \left(15 a^{5} + 4 a^{4} + 26 a^{3} + 4 a^{2} + 30 a + 8\right)\cdot 31^{4} + \left(20 a^{5} + 22 a^{4} + 18 a^{3} + 29 a^{2} + 27 a + 27\right)\cdot 31^{5} + \left(29 a^{5} + a^{4} + 12 a^{3} + 4 a^{2} + 27 a + 16\right)\cdot 31^{6} + \left(22 a^{5} + 2 a^{4} + 20 a^{3} + 10 a^{2} + 22 a + 6\right)\cdot 31^{7} + \left(19 a^{5} + 7 a^{4} + 26 a^{3} + 25 a^{2} + a + 12\right)\cdot 31^{8} + \left(28 a^{5} + 6 a^{4} + 29 a^{3} + 28 a^{2} + 5 a + 10\right)\cdot 31^{9} +O(31^{10})$$ 19*a^5 + 24*a^4 + 3*a^3 + 7*a^2 + 7*a + 9 + (9*a^5 + 22*a^4 + 19*a^3 + 21*a^2 + 28*a + 30)*31 + (22*a^5 + 11*a^4 + 26*a^3 + 6*a + 27)*31^2 + (27*a^5 + 26*a^4 + 22*a^3 + 6*a^2 + 15*a + 23)*31^3 + (15*a^5 + 4*a^4 + 26*a^3 + 4*a^2 + 30*a + 8)*31^4 + (20*a^5 + 22*a^4 + 18*a^3 + 29*a^2 + 27*a + 27)*31^5 + (29*a^5 + a^4 + 12*a^3 + 4*a^2 + 27*a + 16)*31^6 + (22*a^5 + 2*a^4 + 20*a^3 + 10*a^2 + 22*a + 6)*31^7 + (19*a^5 + 7*a^4 + 26*a^3 + 25*a^2 + a + 12)*31^8 + (28*a^5 + 6*a^4 + 29*a^3 + 28*a^2 + 5*a + 10)*31^9+O(31^10) $r_{ 9 }$ $=$ $$18 a^{5} + 2 a^{4} + 27 a^{3} + 7 a^{2} + 24 a + 6 + \left(5 a^{5} + 25 a^{4} + 7 a^{3} + 4 a^{2} + 22 a + 14\right)\cdot 31 + \left(22 a^{5} + 18 a^{4} + 7 a^{3} + 21 a^{2} + 6 a + 6\right)\cdot 31^{2} + \left(29 a^{5} + 20 a^{4} + 7 a^{3} + 26 a^{2} + 7 a + 13\right)\cdot 31^{3} + \left(22 a^{5} + 21 a^{4} + 4 a^{3} + 14 a^{2} + 7 a + 21\right)\cdot 31^{4} + \left(13 a^{5} + 8 a^{4} + 24 a^{3} + 7 a^{2} + 25 a + 12\right)\cdot 31^{5} + \left(26 a^{5} + 25 a^{4} + 16 a^{3} + 29 a^{2} + 18 a + 7\right)\cdot 31^{6} + \left(15 a^{5} + 3 a^{4} + 17 a^{3} + 6 a^{2} + 13 a + 28\right)\cdot 31^{7} + \left(27 a^{5} + 29 a^{4} + 24 a^{3} + 5 a^{2} + 14 a + 15\right)\cdot 31^{8} + \left(13 a^{5} + 26 a^{4} + 4 a^{3} + 25 a^{2} + 16 a + 8\right)\cdot 31^{9} +O(31^{10})$$ 18*a^5 + 2*a^4 + 27*a^3 + 7*a^2 + 24*a + 6 + (5*a^5 + 25*a^4 + 7*a^3 + 4*a^2 + 22*a + 14)*31 + (22*a^5 + 18*a^4 + 7*a^3 + 21*a^2 + 6*a + 6)*31^2 + (29*a^5 + 20*a^4 + 7*a^3 + 26*a^2 + 7*a + 13)*31^3 + (22*a^5 + 21*a^4 + 4*a^3 + 14*a^2 + 7*a + 21)*31^4 + (13*a^5 + 8*a^4 + 24*a^3 + 7*a^2 + 25*a + 12)*31^5 + (26*a^5 + 25*a^4 + 16*a^3 + 29*a^2 + 18*a + 7)*31^6 + (15*a^5 + 3*a^4 + 17*a^3 + 6*a^2 + 13*a + 28)*31^7 + (27*a^5 + 29*a^4 + 24*a^3 + 5*a^2 + 14*a + 15)*31^8 + (13*a^5 + 26*a^4 + 4*a^3 + 25*a^2 + 16*a + 8)*31^9+O(31^10) $r_{ 10 }$ $=$ $$30 a^{5} + 6 a^{4} + 12 a^{3} + 7 a^{2} + 3 a + \left(26 a^{4} + 25 a^{3} + 10 a^{2} + 8 a + 19\right)\cdot 31 + \left(23 a^{5} + 15 a^{4} + 6 a^{3} + 9 a^{2} + 12 a + 2\right)\cdot 31^{2} + \left(11 a^{5} + 11 a^{4} + 18 a^{3} + 9 a^{2} + 26 a + 26\right)\cdot 31^{3} + \left(9 a^{5} + 9 a^{4} + 14 a^{3} + 7 a^{2} + 27 a + 16\right)\cdot 31^{4} + \left(12 a^{5} + 15 a^{4} + a^{3} + 11 a^{2} + 17 a + 19\right)\cdot 31^{5} + \left(12 a^{5} + 12 a^{4} + 15 a^{3} + 9 a^{2} + 2 a + 21\right)\cdot 31^{6} + \left(13 a^{5} + 29 a^{4} + 11 a^{3} + 27 a^{2} + 12 a + 16\right)\cdot 31^{7} + \left(13 a^{5} + 18 a^{4} + 7 a^{3} + 3 a^{2} + 11 a + 29\right)\cdot 31^{8} + \left(4 a^{4} + 4 a^{3} + 19 a^{2} + 19 a + 2\right)\cdot 31^{9} +O(31^{10})$$ 30*a^5 + 6*a^4 + 12*a^3 + 7*a^2 + 3*a + (26*a^4 + 25*a^3 + 10*a^2 + 8*a + 19)*31 + (23*a^5 + 15*a^4 + 6*a^3 + 9*a^2 + 12*a + 2)*31^2 + (11*a^5 + 11*a^4 + 18*a^3 + 9*a^2 + 26*a + 26)*31^3 + (9*a^5 + 9*a^4 + 14*a^3 + 7*a^2 + 27*a + 16)*31^4 + (12*a^5 + 15*a^4 + a^3 + 11*a^2 + 17*a + 19)*31^5 + (12*a^5 + 12*a^4 + 15*a^3 + 9*a^2 + 2*a + 21)*31^6 + (13*a^5 + 29*a^4 + 11*a^3 + 27*a^2 + 12*a + 16)*31^7 + (13*a^5 + 18*a^4 + 7*a^3 + 3*a^2 + 11*a + 29)*31^8 + (4*a^4 + 4*a^3 + 19*a^2 + 19*a + 2)*31^9+O(31^10) $r_{ 11 }$ $=$ $$28 a^{5} + 23 a^{4} + 30 a^{3} + 24 a^{2} + 9 a + 29 + \left(15 a^{5} + 16 a^{4} + 29 a^{3} + 15 a^{2} + 28 a + 19\right)\cdot 31 + \left(27 a^{5} + 23 a^{4} + 16 a^{3} + 14 a^{2} + 12 a + 14\right)\cdot 31^{2} + \left(16 a^{5} + 10 a^{4} + 13 a^{3} + 11 a^{2} + 17 a + 23\right)\cdot 31^{3} + \left(19 a^{5} + 16 a^{4} + 27 a^{3} + 26 a^{2} + 27 a + 22\right)\cdot 31^{4} + \left(a^{5} + 19 a^{4} + 4 a^{3} + 9 a^{2} + 16 a + 20\right)\cdot 31^{5} + \left(28 a^{5} + 13 a^{4} + 21 a^{2} + 16 a + 27\right)\cdot 31^{6} + \left(11 a^{5} + 9 a^{4} + 25 a^{3} + 30 a^{2} + 16 a + 4\right)\cdot 31^{7} + \left(20 a^{5} + 9 a^{4} + 18 a^{3} + 23 a^{2} + 23 a + 4\right)\cdot 31^{8} + \left(26 a^{5} + 22 a^{4} + 4 a^{2} + 14 a + 27\right)\cdot 31^{9} +O(31^{10})$$ 28*a^5 + 23*a^4 + 30*a^3 + 24*a^2 + 9*a + 29 + (15*a^5 + 16*a^4 + 29*a^3 + 15*a^2 + 28*a + 19)*31 + (27*a^5 + 23*a^4 + 16*a^3 + 14*a^2 + 12*a + 14)*31^2 + (16*a^5 + 10*a^4 + 13*a^3 + 11*a^2 + 17*a + 23)*31^3 + (19*a^5 + 16*a^4 + 27*a^3 + 26*a^2 + 27*a + 22)*31^4 + (a^5 + 19*a^4 + 4*a^3 + 9*a^2 + 16*a + 20)*31^5 + (28*a^5 + 13*a^4 + 21*a^2 + 16*a + 27)*31^6 + (11*a^5 + 9*a^4 + 25*a^3 + 30*a^2 + 16*a + 4)*31^7 + (20*a^5 + 9*a^4 + 18*a^3 + 23*a^2 + 23*a + 4)*31^8 + (26*a^5 + 22*a^4 + 4*a^2 + 14*a + 27)*31^9+O(31^10) $r_{ 12 }$ $=$ $$28 a^{5} + 30 a^{4} + 26 a^{3} + 24 a^{2} + 9 a + 14 + \left(13 a^{5} + 30 a^{4} + 12 a^{3} + 26 a^{2} + 25 a + 21\right)\cdot 31 + \left(3 a^{5} + 19 a^{4} + a^{3} + 22 a^{2} + 16 a + 9\right)\cdot 31^{2} + \left(12 a^{5} + 2 a^{4} + 21 a^{3} + 6 a^{2} + 20 a + 7\right)\cdot 31^{3} + \left(14 a^{5} + 13 a^{4} + 13 a^{3} + 20 a^{2} + 21 a + 19\right)\cdot 31^{4} + \left(4 a^{5} + 4 a^{4} + 16 a^{3} + 2 a^{2} + 2 a + 29\right)\cdot 31^{5} + \left(a^{5} + 17 a^{4} + 24 a^{3} + 29 a^{2} + 19 a + 13\right)\cdot 31^{6} + \left(10 a^{5} + 29 a^{4} + 19 a^{3} + 10 a^{2} + 4 a + 2\right)\cdot 31^{7} + \left(15 a^{5} + 16 a^{4} + 11 a^{3} + 16 a^{2} + 12 a + 20\right)\cdot 31^{8} + \left(24 a^{5} + 14 a^{4} + 3 a^{3} + 22 a^{2} + 6 a + 28\right)\cdot 31^{9} +O(31^{10})$$ 28*a^5 + 30*a^4 + 26*a^3 + 24*a^2 + 9*a + 14 + (13*a^5 + 30*a^4 + 12*a^3 + 26*a^2 + 25*a + 21)*31 + (3*a^5 + 19*a^4 + a^3 + 22*a^2 + 16*a + 9)*31^2 + (12*a^5 + 2*a^4 + 21*a^3 + 6*a^2 + 20*a + 7)*31^3 + (14*a^5 + 13*a^4 + 13*a^3 + 20*a^2 + 21*a + 19)*31^4 + (4*a^5 + 4*a^4 + 16*a^3 + 2*a^2 + 2*a + 29)*31^5 + (a^5 + 17*a^4 + 24*a^3 + 29*a^2 + 19*a + 13)*31^6 + (10*a^5 + 29*a^4 + 19*a^3 + 10*a^2 + 4*a + 2)*31^7 + (15*a^5 + 16*a^4 + 11*a^3 + 16*a^2 + 12*a + 20)*31^8 + (24*a^5 + 14*a^4 + 3*a^3 + 22*a^2 + 6*a + 28)*31^9+O(31^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.