# Properties

 Label 11.125...000.24t2949.a.a Dimension $11$ Group $\PGL(2,11)$ Conductor $1.255\times 10^{21}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $11$ Group: $\PGL(2,11)$ Conductor: $$125\!\cdots\!000$$$$\medspace = 2^{10} \cdot 5^{8} \cdot 11^{12}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 12.2.114124668244400000000.1 Galois orbit size: $1$ Smallest permutation container: 24T2949 Parity: even Determinant: 1.1.1t1.a.a Projective image: $\PSL(2,11).C_2$ Projective stem field: Galois closure of 12.2.114124668244400000000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - 44 x^{9} + 44 x^{8} - 110 x^{7} + 682 x^{6} - 1364 x^{5} + 1452 x^{4} - 3982 x^{3} + 10406 x^{2} - 11826 x + 4972$$ x^12 - 44*x^9 + 44*x^8 - 110*x^7 + 682*x^6 - 1364*x^5 + 1452*x^4 - 3982*x^3 + 10406*x^2 - 11826*x + 4972 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{6} + x^{4} + 9x^{3} + 9x^{2} + x + 5$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.