# Properties

 Label 1.147.14t1.a.e Dimension $1$ Group $C_{14}$ Conductor $147$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{14}$ Conductor: $$147$$$$\medspace = 3 \cdot 7^{2}$$ Artin field: Galois closure of 14.0.418988153029298748294987.1 Galois orbit size: $6$ Smallest permutation container: $C_{14}$ Parity: odd Dirichlet character: $$\chi_{147}(113,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} - 8701 x^{5} + 18151 x^{4} - 8246 x^{3} + 17920 x^{2} - 8148 x + 9409$$ x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409 .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{7} + 7x + 35$$

Roots:
 $r_{ 1 }$ $=$ $$4 a^{6} + 34 a^{5} + 36 a^{3} + 5 a^{2} + 2 a + 24 + \left(15 a^{6} + 23 a^{5} + 25 a^{4} + 9 a^{3} + 4 a^{2} + 30 a + 16\right)\cdot 37 + \left(12 a^{6} + 21 a^{5} + 9 a^{4} + 10 a^{3} + 33 a^{2} + 15 a\right)\cdot 37^{2} + \left(35 a^{6} + 16 a^{5} + 28 a^{4} + 7 a^{3} + 13 a^{2} + 2 a + 27\right)\cdot 37^{3} + \left(10 a^{6} + 27 a^{5} + 12 a^{3} + 31 a^{2} + 27 a + 28\right)\cdot 37^{4} + \left(12 a^{6} + 27 a^{5} + 21 a^{4} + 21 a^{3} + 23 a^{2} + 36\right)\cdot 37^{5} + \left(8 a^{6} + 6 a^{5} + 27 a^{4} + 4 a^{3} + 15 a^{2} + 5 a + 12\right)\cdot 37^{6} + \left(2 a^{6} + 36 a^{5} + 31 a^{4} + 4 a^{3} + 9 a^{2} + 9 a + 13\right)\cdot 37^{7} + \left(34 a^{6} + 3 a^{5} + 6 a^{4} + 4 a^{3} + 28 a^{2} + 11 a + 19\right)\cdot 37^{8} +O(37^{9})$$ 4*a^6 + 34*a^5 + 36*a^3 + 5*a^2 + 2*a + 24 + (15*a^6 + 23*a^5 + 25*a^4 + 9*a^3 + 4*a^2 + 30*a + 16)*37 + (12*a^6 + 21*a^5 + 9*a^4 + 10*a^3 + 33*a^2 + 15*a)*37^2 + (35*a^6 + 16*a^5 + 28*a^4 + 7*a^3 + 13*a^2 + 2*a + 27)*37^3 + (10*a^6 + 27*a^5 + 12*a^3 + 31*a^2 + 27*a + 28)*37^4 + (12*a^6 + 27*a^5 + 21*a^4 + 21*a^3 + 23*a^2 + 36)*37^5 + (8*a^6 + 6*a^5 + 27*a^4 + 4*a^3 + 15*a^2 + 5*a + 12)*37^6 + (2*a^6 + 36*a^5 + 31*a^4 + 4*a^3 + 9*a^2 + 9*a + 13)*37^7 + (34*a^6 + 3*a^5 + 6*a^4 + 4*a^3 + 28*a^2 + 11*a + 19)*37^8+O(37^9) $r_{ 2 }$ $=$ $$6 a^{6} + 11 a^{5} + 31 a^{4} + 31 a^{3} + 4 a + 36 + \left(31 a^{6} + 32 a^{5} + 21 a^{4} + 22 a^{3} + 30 a^{2} + 2 a + 1\right)\cdot 37 + \left(17 a^{6} + a^{5} + 35 a^{4} + 6 a^{3} + 33 a^{2} + 32 a + 33\right)\cdot 37^{2} + \left(6 a^{6} + 16 a^{5} + 5 a^{4} + 20 a^{3} + a^{2} + 31 a + 1\right)\cdot 37^{3} + \left(30 a^{6} + 36 a^{5} + 20 a^{4} + 12 a^{3} + 4 a^{2} + 34 a + 33\right)\cdot 37^{4} + \left(21 a^{6} + 31 a^{5} + 21 a^{4} + 34 a^{3} + 12 a^{2} + 20 a + 19\right)\cdot 37^{5} + \left(20 a^{6} + 4 a^{5} + 19 a^{4} + 36 a^{3} + 5 a^{2} + 2 a + 12\right)\cdot 37^{6} + \left(5 a^{6} + 28 a^{5} + 13 a^{4} + 3 a^{3} + 13 a^{2} + 27 a + 33\right)\cdot 37^{7} + \left(26 a^{6} + 21 a^{5} + 15 a^{4} + 33 a^{3} + 29 a^{2} + 30 a + 8\right)\cdot 37^{8} +O(37^{9})$$ 6*a^6 + 11*a^5 + 31*a^4 + 31*a^3 + 4*a + 36 + (31*a^6 + 32*a^5 + 21*a^4 + 22*a^3 + 30*a^2 + 2*a + 1)*37 + (17*a^6 + a^5 + 35*a^4 + 6*a^3 + 33*a^2 + 32*a + 33)*37^2 + (6*a^6 + 16*a^5 + 5*a^4 + 20*a^3 + a^2 + 31*a + 1)*37^3 + (30*a^6 + 36*a^5 + 20*a^4 + 12*a^3 + 4*a^2 + 34*a + 33)*37^4 + (21*a^6 + 31*a^5 + 21*a^4 + 34*a^3 + 12*a^2 + 20*a + 19)*37^5 + (20*a^6 + 4*a^5 + 19*a^4 + 36*a^3 + 5*a^2 + 2*a + 12)*37^6 + (5*a^6 + 28*a^5 + 13*a^4 + 3*a^3 + 13*a^2 + 27*a + 33)*37^7 + (26*a^6 + 21*a^5 + 15*a^4 + 33*a^3 + 29*a^2 + 30*a + 8)*37^8+O(37^9) $r_{ 3 }$ $=$ $$6 a^{6} + 34 a^{5} + 21 a^{4} + 21 a^{3} + 10 a + 36 + \left(21 a^{6} + 8 a^{5} + 12 a^{4} + 36 a^{3} + 15 a^{2} + 25 a + 15\right)\cdot 37 + \left(17 a^{6} + 13 a^{5} + 35 a^{4} + 30 a^{3} + 7 a^{2} + 11 a + 31\right)\cdot 37^{2} + \left(16 a^{6} + a^{5} + 5 a^{4} + 32 a^{3} + 23 a^{2} + 3 a + 24\right)\cdot 37^{3} + \left(36 a^{6} + 10 a^{5} + 17 a^{4} + 14 a^{3} + 26 a^{2} + 18 a + 33\right)\cdot 37^{4} + \left(a^{6} + 32 a^{5} + 24 a^{4} + 3 a^{3} + 11 a^{2} + 11 a + 11\right)\cdot 37^{5} + \left(23 a^{6} + 19 a^{5} + 12 a^{4} + 34 a^{3} + 21 a^{2} + 27 a + 27\right)\cdot 37^{6} + \left(4 a^{6} + 20 a^{5} + 16 a^{4} + 26 a^{3} + 6 a^{2} + 27 a + 27\right)\cdot 37^{7} + \left(30 a^{6} + 32 a^{5} + 17 a^{4} + 4 a^{3} + 9 a^{2} + 25 a + 32\right)\cdot 37^{8} +O(37^{9})$$ 6*a^6 + 34*a^5 + 21*a^4 + 21*a^3 + 10*a + 36 + (21*a^6 + 8*a^5 + 12*a^4 + 36*a^3 + 15*a^2 + 25*a + 15)*37 + (17*a^6 + 13*a^5 + 35*a^4 + 30*a^3 + 7*a^2 + 11*a + 31)*37^2 + (16*a^6 + a^5 + 5*a^4 + 32*a^3 + 23*a^2 + 3*a + 24)*37^3 + (36*a^6 + 10*a^5 + 17*a^4 + 14*a^3 + 26*a^2 + 18*a + 33)*37^4 + (a^6 + 32*a^5 + 24*a^4 + 3*a^3 + 11*a^2 + 11*a + 11)*37^5 + (23*a^6 + 19*a^5 + 12*a^4 + 34*a^3 + 21*a^2 + 27*a + 27)*37^6 + (4*a^6 + 20*a^5 + 16*a^4 + 26*a^3 + 6*a^2 + 27*a + 27)*37^7 + (30*a^6 + 32*a^5 + 17*a^4 + 4*a^3 + 9*a^2 + 25*a + 32)*37^8+O(37^9) $r_{ 4 }$ $=$ $$9 a^{6} + 30 a^{5} + 28 a^{4} + 33 a^{3} + 23 a^{2} + 29 a + 17 + \left(a^{5} + 33 a^{4} + 15 a^{3} + 18 a^{2} + 11 a + 1\right)\cdot 37 + \left(9 a^{6} + 16 a^{5} + 36 a^{4} + 26 a^{3} + 21 a^{2} + 29 a + 17\right)\cdot 37^{2} + \left(7 a^{6} + 24 a^{5} + 8 a^{4} + 6 a^{3} + 16 a^{2} + 30 a + 6\right)\cdot 37^{3} + \left(6 a^{6} + 14 a^{5} + 8 a^{4} + 9 a^{3} + 6 a^{2} + 25 a\right)\cdot 37^{4} + \left(18 a^{6} + 26 a^{5} + 21 a^{4} + 24 a^{3} + 26 a^{2} + 12 a + 35\right)\cdot 37^{5} + \left(36 a^{5} + 27 a^{4} + 2 a^{3} + 2 a^{2} + 24 a + 2\right)\cdot 37^{6} + \left(15 a^{6} + 22 a^{5} + 7 a^{4} + 12 a^{3} + 29 a^{2} + 22 a + 16\right)\cdot 37^{7} + \left(7 a^{6} + 8 a^{5} + 9 a^{4} + 4 a^{3} + 12 a^{2} + 22 a + 7\right)\cdot 37^{8} +O(37^{9})$$ 9*a^6 + 30*a^5 + 28*a^4 + 33*a^3 + 23*a^2 + 29*a + 17 + (a^5 + 33*a^4 + 15*a^3 + 18*a^2 + 11*a + 1)*37 + (9*a^6 + 16*a^5 + 36*a^4 + 26*a^3 + 21*a^2 + 29*a + 17)*37^2 + (7*a^6 + 24*a^5 + 8*a^4 + 6*a^3 + 16*a^2 + 30*a + 6)*37^3 + (6*a^6 + 14*a^5 + 8*a^4 + 9*a^3 + 6*a^2 + 25*a)*37^4 + (18*a^6 + 26*a^5 + 21*a^4 + 24*a^3 + 26*a^2 + 12*a + 35)*37^5 + (36*a^5 + 27*a^4 + 2*a^3 + 2*a^2 + 24*a + 2)*37^6 + (15*a^6 + 22*a^5 + 7*a^4 + 12*a^3 + 29*a^2 + 22*a + 16)*37^7 + (7*a^6 + 8*a^5 + 9*a^4 + 4*a^3 + 12*a^2 + 22*a + 7)*37^8+O(37^9) $r_{ 5 }$ $=$ $$9 a^{6} + 33 a^{5} + 18 a^{4} + 16 a^{3} + 13 a^{2} + 31 a + 17 + \left(32 a^{6} + 8 a^{5} + 31 a^{4} + 7 a^{3} + 23 a^{2} + 10 a + 8\right)\cdot 37 + \left(36 a^{6} + 7 a^{5} + 28 a^{4} + 18 a^{3} + 12 a^{2} + 19 a + 36\right)\cdot 37^{2} + \left(32 a^{6} + 16 a^{4} + 5 a^{3} + 36 a^{2} + 12\right)\cdot 37^{3} + \left(10 a^{6} + 9 a^{5} + 34 a^{4} + 13 a^{3} + 31 a^{2} + 8 a + 28\right)\cdot 37^{4} + \left(34 a^{6} + 28 a^{5} + 36 a^{4} + 15 a^{3} + 26 a^{2} + 10 a + 20\right)\cdot 37^{5} + \left(2 a^{6} + 28 a^{5} + 6 a^{4} + 28 a^{3} + 8 a^{2} + 11 a + 17\right)\cdot 37^{6} + \left(23 a^{6} + a^{5} + 34 a^{4} + 7 a^{3} + 19 a^{2} + 9 a + 27\right)\cdot 37^{7} + \left(9 a^{6} + 14 a^{5} + 22 a^{4} + 4 a^{3} + 18 a^{2} + 30 a + 20\right)\cdot 37^{8} +O(37^{9})$$ 9*a^6 + 33*a^5 + 18*a^4 + 16*a^3 + 13*a^2 + 31*a + 17 + (32*a^6 + 8*a^5 + 31*a^4 + 7*a^3 + 23*a^2 + 10*a + 8)*37 + (36*a^6 + 7*a^5 + 28*a^4 + 18*a^3 + 12*a^2 + 19*a + 36)*37^2 + (32*a^6 + 16*a^4 + 5*a^3 + 36*a^2 + 12)*37^3 + (10*a^6 + 9*a^5 + 34*a^4 + 13*a^3 + 31*a^2 + 8*a + 28)*37^4 + (34*a^6 + 28*a^5 + 36*a^4 + 15*a^3 + 26*a^2 + 10*a + 20)*37^5 + (2*a^6 + 28*a^5 + 6*a^4 + 28*a^3 + 8*a^2 + 11*a + 17)*37^6 + (23*a^6 + a^5 + 34*a^4 + 7*a^3 + 19*a^2 + 9*a + 27)*37^7 + (9*a^6 + 14*a^5 + 22*a^4 + 4*a^3 + 18*a^2 + 30*a + 20)*37^8+O(37^9) $r_{ 6 }$ $=$ $$12 a^{6} + 7 a^{5} + 24 a^{4} + 9 a^{3} + 5 a^{2} + 29 a + 35 + \left(11 a^{6} + 23 a^{5} + 15 a^{4} + 26 a^{3} + 25 a^{2} + 6 a + 30\right)\cdot 37 + \left(15 a^{6} + 22 a^{5} + 33 a^{4} + 4 a^{3} + 31 a^{2} + 12 a + 17\right)\cdot 37^{2} + \left(10 a^{6} + 14 a^{5} + 31 a^{4} + 21 a^{3} + 17 a^{2} + 18 a + 25\right)\cdot 37^{3} + \left(17 a^{6} + 32 a^{5} + 30 a^{4} + 35 a^{3} + 21 a^{2} + 5 a + 29\right)\cdot 37^{4} + \left(a^{6} + 17 a^{5} + 15 a^{4} + 16 a^{3} + 27 a^{2} + 30 a + 8\right)\cdot 37^{5} + \left(23 a^{6} + 4 a^{5} + 12 a^{4} + 27 a^{3} + 28 a^{2} + 8 a + 27\right)\cdot 37^{6} + \left(28 a^{6} + 17 a^{5} + 21 a^{4} + a^{3} + 16 a^{2} + 26 a + 23\right)\cdot 37^{7} + \left(7 a^{6} + 3 a^{5} + 6 a^{4} + 28 a^{3} + 4 a^{2} + 16 a + 9\right)\cdot 37^{8} +O(37^{9})$$ 12*a^6 + 7*a^5 + 24*a^4 + 9*a^3 + 5*a^2 + 29*a + 35 + (11*a^6 + 23*a^5 + 15*a^4 + 26*a^3 + 25*a^2 + 6*a + 30)*37 + (15*a^6 + 22*a^5 + 33*a^4 + 4*a^3 + 31*a^2 + 12*a + 17)*37^2 + (10*a^6 + 14*a^5 + 31*a^4 + 21*a^3 + 17*a^2 + 18*a + 25)*37^3 + (17*a^6 + 32*a^5 + 30*a^4 + 35*a^3 + 21*a^2 + 5*a + 29)*37^4 + (a^6 + 17*a^5 + 15*a^4 + 16*a^3 + 27*a^2 + 30*a + 8)*37^5 + (23*a^6 + 4*a^5 + 12*a^4 + 27*a^3 + 28*a^2 + 8*a + 27)*37^6 + (28*a^6 + 17*a^5 + 21*a^4 + a^3 + 16*a^2 + 26*a + 23)*37^7 + (7*a^6 + 3*a^5 + 6*a^4 + 28*a^3 + 4*a^2 + 16*a + 9)*37^8+O(37^9) $r_{ 7 }$ $=$ $$14 a^{6} + 21 a^{5} + 11 a^{4} + 22 a^{3} + 30 a^{2} + 30 a + 10 + \left(29 a^{6} + 19 a^{5} + 15 a^{4} + 30 a^{3} + 22 a^{2} + 34 a + 28\right)\cdot 37 + \left(12 a^{5} + 24 a^{4} + 36 a^{3} + 28 a^{2} + 25 a + 4\right)\cdot 37^{2} + \left(22 a^{6} + 30 a^{5} + 22 a^{4} + 11 a^{3} + 31 a^{2} + 6 a + 21\right)\cdot 37^{3} + \left(15 a^{6} + 9 a^{5} + 27 a^{4} + 26 a^{3} + 11 a^{2} + 29 a + 19\right)\cdot 37^{4} + \left(5 a^{6} + 31 a^{5} + 7 a^{4} + 32 a^{3} + 28 a^{2} + 13 a + 32\right)\cdot 37^{5} + \left(9 a^{6} + 32 a^{5} + 7 a^{4} + 32 a^{3} + 3 a^{2} + 8 a + 17\right)\cdot 37^{6} + \left(31 a^{6} + 27 a^{5} + 24 a^{4} + 2 a^{3} + 31 a^{2} + 19 a + 2\right)\cdot 37^{7} + \left(11 a^{6} + 23 a^{5} + 22 a^{4} + 19 a^{3} + 36 a^{2} + a + 34\right)\cdot 37^{8} +O(37^{9})$$ 14*a^6 + 21*a^5 + 11*a^4 + 22*a^3 + 30*a^2 + 30*a + 10 + (29*a^6 + 19*a^5 + 15*a^4 + 30*a^3 + 22*a^2 + 34*a + 28)*37 + (12*a^5 + 24*a^4 + 36*a^3 + 28*a^2 + 25*a + 4)*37^2 + (22*a^6 + 30*a^5 + 22*a^4 + 11*a^3 + 31*a^2 + 6*a + 21)*37^3 + (15*a^6 + 9*a^5 + 27*a^4 + 26*a^3 + 11*a^2 + 29*a + 19)*37^4 + (5*a^6 + 31*a^5 + 7*a^4 + 32*a^3 + 28*a^2 + 13*a + 32)*37^5 + (9*a^6 + 32*a^5 + 7*a^4 + 32*a^3 + 3*a^2 + 8*a + 17)*37^6 + (31*a^6 + 27*a^5 + 24*a^4 + 2*a^3 + 31*a^2 + 19*a + 2)*37^7 + (11*a^6 + 23*a^5 + 22*a^4 + 19*a^3 + 36*a^2 + a + 34)*37^8+O(37^9) $r_{ 8 }$ $=$ $$16 a^{6} + 4 a^{5} + 21 a^{4} + 34 a^{3} + 8 a^{2} + 31 a + 22 + \left(6 a^{6} + 19 a^{5} + 3 a^{2} + 23 a + 1\right)\cdot 37 + \left(33 a^{6} + 16 a^{5} + 30 a^{4} + 13 a^{3} + 20 a^{2} + 36 a + 14\right)\cdot 37^{2} + \left(20 a^{6} + 2 a^{5} + 28 a^{4} + 6 a^{3} + 34 a^{2} + 22 a + 14\right)\cdot 37^{3} + \left(10 a^{6} + 14 a^{5} + 36 a^{4} + a^{3} + 12 a^{2} + 12 a + 26\right)\cdot 37^{4} + \left(12 a^{6} + 19 a^{5} + 6 a^{4} + 31 a^{3} + 27 a^{2} + 33 a + 36\right)\cdot 37^{5} + \left(13 a^{6} + 12 a^{5} + 36 a^{4} + 31 a^{3} + 18 a^{2} + 33 a + 5\right)\cdot 37^{6} + \left(26 a^{6} + 18 a^{5} + 6 a^{4} + 34 a^{3} + 13 a^{2} + 31 a + 10\right)\cdot 37^{7} + \left(29 a^{6} + 4 a^{5} + 3 a^{4} + 22 a^{3} + 29 a^{2} + 32 a + 30\right)\cdot 37^{8} +O(37^{9})$$ 16*a^6 + 4*a^5 + 21*a^4 + 34*a^3 + 8*a^2 + 31*a + 22 + (6*a^6 + 19*a^5 + 3*a^2 + 23*a + 1)*37 + (33*a^6 + 16*a^5 + 30*a^4 + 13*a^3 + 20*a^2 + 36*a + 14)*37^2 + (20*a^6 + 2*a^5 + 28*a^4 + 6*a^3 + 34*a^2 + 22*a + 14)*37^3 + (10*a^6 + 14*a^5 + 36*a^4 + a^3 + 12*a^2 + 12*a + 26)*37^4 + (12*a^6 + 19*a^5 + 6*a^4 + 31*a^3 + 27*a^2 + 33*a + 36)*37^5 + (13*a^6 + 12*a^5 + 36*a^4 + 31*a^3 + 18*a^2 + 33*a + 5)*37^6 + (26*a^6 + 18*a^5 + 6*a^4 + 34*a^3 + 13*a^2 + 31*a + 10)*37^7 + (29*a^6 + 4*a^5 + 3*a^4 + 22*a^3 + 29*a^2 + 32*a + 30)*37^8+O(37^9) $r_{ 9 }$ $=$ $$17 a^{6} + 5 a^{5} + 17 a^{4} + 26 a^{3} + 24 a^{2} + 21 a + 28 + \left(26 a^{6} + 13 a^{5} + 35 a^{4} + 8 a^{3} + 17 a^{2} + 7 a + 10\right)\cdot 37 + \left(7 a^{6} + a^{5} + 33 a^{4} + 34 a^{3} + 25 a^{2} + 15 a + 9\right)\cdot 37^{2} + \left(13 a^{6} + a^{5} + 3 a^{4} + 19 a^{3} + 19 a^{2} + 7 a + 5\right)\cdot 37^{3} + \left(20 a^{6} + 28 a^{5} + 20 a^{4} + a^{3} + 18 a^{2} + 20 a + 11\right)\cdot 37^{4} + \left(26 a^{6} + 26 a^{5} + 27 a^{4} + 21 a^{3} + 11 a^{2} + 25 a + 11\right)\cdot 37^{5} + \left(26 a^{6} + 33 a^{5} + 14 a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 12\right)\cdot 37^{6} + \left(33 a^{6} + 4 a^{5} + 7 a^{4} + 32 a^{2} + 12 a + 17\right)\cdot 37^{7} + \left(32 a^{6} + 11 a^{5} + 12 a^{4} + 10 a^{3} + 6 a^{2} + 14 a + 12\right)\cdot 37^{8} +O(37^{9})$$ 17*a^6 + 5*a^5 + 17*a^4 + 26*a^3 + 24*a^2 + 21*a + 28 + (26*a^6 + 13*a^5 + 35*a^4 + 8*a^3 + 17*a^2 + 7*a + 10)*37 + (7*a^6 + a^5 + 33*a^4 + 34*a^3 + 25*a^2 + 15*a + 9)*37^2 + (13*a^6 + a^5 + 3*a^4 + 19*a^3 + 19*a^2 + 7*a + 5)*37^3 + (20*a^6 + 28*a^5 + 20*a^4 + a^3 + 18*a^2 + 20*a + 11)*37^4 + (26*a^6 + 26*a^5 + 27*a^4 + 21*a^3 + 11*a^2 + 25*a + 11)*37^5 + (26*a^6 + 33*a^5 + 14*a^4 + 10*a^3 + 15*a^2 + 5*a + 12)*37^6 + (33*a^6 + 4*a^5 + 7*a^4 + 32*a^2 + 12*a + 17)*37^7 + (32*a^6 + 11*a^5 + 12*a^4 + 10*a^3 + 6*a^2 + 14*a + 12)*37^8+O(37^9) $r_{ 10 }$ $=$ $$22 a^{6} + 13 a^{5} + 22 a^{4} + a^{3} + 18 a^{2} + 25 a + 21 + \left(29 a^{6} + 14 a^{5} + 8 a^{4} + 4 a^{3} + 14 a^{2} + 35 a + 29\right)\cdot 37 + \left(a^{6} + 12 a^{5} + 9 a^{4} + 17 a^{3} + 28 a^{2} + 5 a + 10\right)\cdot 37^{2} + \left(18 a^{6} + 29 a^{5} + 7 a^{4} + 7 a^{3} + a^{2} + 26 a + 34\right)\cdot 37^{3} + \left(12 a^{5} + 16 a^{4} + 33 a^{3} + 18 a^{2} + 24 a + 2\right)\cdot 37^{4} + \left(13 a^{6} + 3 a^{5} + 14 a^{4} + 24 a^{2} + 14 a + 4\right)\cdot 37^{5} + \left(2 a^{6} + 9 a^{5} + 6 a^{4} + 21 a^{3} + 13 a^{2} + 9 a + 14\right)\cdot 37^{6} + \left(12 a^{6} + 21 a^{5} + 30 a^{4} + 11 a^{3} + 27 a^{2} + 15 a + 35\right)\cdot 37^{7} + \left(13 a^{6} + 25 a^{5} + 21 a^{4} + 18 a^{3} + 6 a^{2} + 8 a + 5\right)\cdot 37^{8} +O(37^{9})$$ 22*a^6 + 13*a^5 + 22*a^4 + a^3 + 18*a^2 + 25*a + 21 + (29*a^6 + 14*a^5 + 8*a^4 + 4*a^3 + 14*a^2 + 35*a + 29)*37 + (a^6 + 12*a^5 + 9*a^4 + 17*a^3 + 28*a^2 + 5*a + 10)*37^2 + (18*a^6 + 29*a^5 + 7*a^4 + 7*a^3 + a^2 + 26*a + 34)*37^3 + (12*a^5 + 16*a^4 + 33*a^3 + 18*a^2 + 24*a + 2)*37^4 + (13*a^6 + 3*a^5 + 14*a^4 + 24*a^2 + 14*a + 4)*37^5 + (2*a^6 + 9*a^5 + 6*a^4 + 21*a^3 + 13*a^2 + 9*a + 14)*37^6 + (12*a^6 + 21*a^5 + 30*a^4 + 11*a^3 + 27*a^2 + 15*a + 35)*37^7 + (13*a^6 + 25*a^5 + 21*a^4 + 18*a^3 + 6*a^2 + 8*a + 5)*37^8+O(37^9) $r_{ 11 }$ $=$ $$23 a^{6} + 7 a^{5} + 25 a^{4} + 25 a^{3} + 26 a + 27 + \left(4 a^{6} + 26 a^{5} + 11 a^{4} + 29 a^{3} + 2 a^{2} + 18 a + 27\right)\cdot 37 + \left(36 a^{6} + 19 a^{5} + 16 a^{4} + 32 a^{3} + 19 a^{2} + 19 a + 31\right)\cdot 37^{2} + \left(18 a^{6} + a^{5} + 9 a^{4} + 33 a^{3} + 11 a + 2\right)\cdot 37^{3} + \left(15 a^{6} + 2 a^{5} + 25 a^{4} + 4 a^{3} + 4 a^{2} + 27 a + 19\right)\cdot 37^{4} + \left(17 a^{6} + 19 a^{5} + 18 a^{4} + 13 a^{3} + 29 a^{2} + 16 a + 30\right)\cdot 37^{5} + \left(24 a^{6} + 25 a^{5} + 23 a^{4} + 2 a^{3} + 8 a^{2} + 34 a + 35\right)\cdot 37^{6} + \left(7 a^{6} + 17 a^{5} + 18 a^{4} + 29 a^{3} + 24 a^{2} + 11 a + 8\right)\cdot 37^{7} + \left(25 a^{6} + 22 a^{5} + 6 a^{4} + 25 a^{3} + 3 a^{2} + 7 a + 3\right)\cdot 37^{8} +O(37^{9})$$ 23*a^6 + 7*a^5 + 25*a^4 + 25*a^3 + 26*a + 27 + (4*a^6 + 26*a^5 + 11*a^4 + 29*a^3 + 2*a^2 + 18*a + 27)*37 + (36*a^6 + 19*a^5 + 16*a^4 + 32*a^3 + 19*a^2 + 19*a + 31)*37^2 + (18*a^6 + a^5 + 9*a^4 + 33*a^3 + 11*a + 2)*37^3 + (15*a^6 + 2*a^5 + 25*a^4 + 4*a^3 + 4*a^2 + 27*a + 19)*37^4 + (17*a^6 + 19*a^5 + 18*a^4 + 13*a^3 + 29*a^2 + 16*a + 30)*37^5 + (24*a^6 + 25*a^5 + 23*a^4 + 2*a^3 + 8*a^2 + 34*a + 35)*37^6 + (7*a^6 + 17*a^5 + 18*a^4 + 29*a^3 + 24*a^2 + 11*a + 8)*37^7 + (25*a^6 + 22*a^5 + 6*a^4 + 25*a^3 + 3*a^2 + 7*a + 3)*37^8+O(37^9) $r_{ 12 }$ $=$ $$23 a^{6} + 36 a^{5} + 14 a^{4} + 14 a^{3} + 3 a + 27 + \left(30 a^{6} + 35 a^{5} + 18 a^{4} + 28 a^{3} + 4 a^{2} + 31 a + 35\right)\cdot 37 + \left(26 a^{6} + 20 a^{5} + 29 a^{4} + 19 a^{3} + 6 a^{2} + 24 a + 12\right)\cdot 37^{2} + \left(36 a^{6} + 30 a^{5} + 34 a^{4} + a^{3} + 29 a^{2} + 3 a + 35\right)\cdot 37^{3} + \left(9 a^{6} + 35 a^{5} + 6 a^{4} + 20 a^{3} + 19 a + 22\right)\cdot 37^{4} + \left(16 a^{6} + 18 a^{5} + 5 a^{4} + 33 a^{3} + 25 a^{2} + 21 a + 23\right)\cdot 37^{5} + \left(13 a^{6} + 32 a^{5} + 3 a^{4} + 26 a^{3} + 4 a^{2} + 8 a + 6\right)\cdot 37^{6} + \left(8 a^{6} + 24 a^{5} + 2 a^{4} + 20 a^{3} + 23 a^{2} + 14 a + 13\right)\cdot 37^{7} + \left(24 a^{6} + 16 a^{5} + 27 a^{4} + 16 a^{3} + 24 a^{2} + 19 a + 34\right)\cdot 37^{8} +O(37^{9})$$ 23*a^6 + 36*a^5 + 14*a^4 + 14*a^3 + 3*a + 27 + (30*a^6 + 35*a^5 + 18*a^4 + 28*a^3 + 4*a^2 + 31*a + 35)*37 + (26*a^6 + 20*a^5 + 29*a^4 + 19*a^3 + 6*a^2 + 24*a + 12)*37^2 + (36*a^6 + 30*a^5 + 34*a^4 + a^3 + 29*a^2 + 3*a + 35)*37^3 + (9*a^6 + 35*a^5 + 6*a^4 + 20*a^3 + 19*a + 22)*37^4 + (16*a^6 + 18*a^5 + 5*a^4 + 33*a^3 + 25*a^2 + 21*a + 23)*37^5 + (13*a^6 + 32*a^5 + 3*a^4 + 26*a^3 + 4*a^2 + 8*a + 6)*37^6 + (8*a^6 + 24*a^5 + 2*a^4 + 20*a^3 + 23*a^2 + 14*a + 13)*37^7 + (24*a^6 + 16*a^5 + 27*a^4 + 16*a^3 + 24*a^2 + 19*a + 34)*37^8+O(37^9) $r_{ 13 }$ $=$ $$30 a^{6} + 33 a^{5} + 11 a^{3} + 19 a^{2} + 15 a + 32 + \left(8 a^{6} + 21 a^{5} + 21 a^{4} + 22 a^{3} + 34 a^{2} + 34 a + 15\right)\cdot 37 + \left(34 a^{6} + 3 a^{5} + 34 a^{4} + 36 a^{3} + 10 a^{2} + 30 a + 20\right)\cdot 37^{2} + \left(3 a^{5} + 19 a^{4} + 25 a^{3} + 24 a^{2} + 28 a + 5\right)\cdot 37^{3} + \left(13 a^{6} + 15 a^{5} + 13 a^{4} + 22 a^{2} + 12 a + 4\right)\cdot 37^{4} + \left(13 a^{6} + 17 a^{5} + 24 a^{4} + 17 a^{3} + 29 a^{2} + 18 a + 6\right)\cdot 37^{5} + \left(20 a^{6} + 18 a^{5} + 4 a^{4} + 17 a^{3} + 35 a^{2} + 15 a + 11\right)\cdot 37^{6} + \left(30 a^{6} + 30 a^{5} + a^{4} + 19 a^{3} + 23 a^{2} + 35\right)\cdot 37^{7} + \left(26 a^{6} + 4 a^{5} + 25 a^{4} + 6 a^{3} + 36 a^{2} + 25 a + 12\right)\cdot 37^{8} +O(37^{9})$$ 30*a^6 + 33*a^5 + 11*a^3 + 19*a^2 + 15*a + 32 + (8*a^6 + 21*a^5 + 21*a^4 + 22*a^3 + 34*a^2 + 34*a + 15)*37 + (34*a^6 + 3*a^5 + 34*a^4 + 36*a^3 + 10*a^2 + 30*a + 20)*37^2 + (3*a^5 + 19*a^4 + 25*a^3 + 24*a^2 + 28*a + 5)*37^3 + (13*a^6 + 15*a^5 + 13*a^4 + 22*a^2 + 12*a + 4)*37^4 + (13*a^6 + 17*a^5 + 24*a^4 + 17*a^3 + 29*a^2 + 18*a + 6)*37^5 + (20*a^6 + 18*a^5 + 4*a^4 + 17*a^3 + 35*a^2 + 15*a + 11)*37^6 + (30*a^6 + 30*a^5 + a^4 + 19*a^3 + 23*a^2 + 35)*37^7 + (26*a^6 + 4*a^5 + 25*a^4 + 6*a^3 + 36*a^2 + 25*a + 12)*37^8+O(37^9) $r_{ 14 }$ $=$ $$31 a^{6} + 28 a^{5} + 27 a^{4} + 17 a^{3} + 3 a^{2} + 3 a + 1 + \left(11 a^{6} + 9 a^{5} + 7 a^{4} + 15 a^{3} + 7 a^{2} + 23 a + 34\right)\cdot 37 + \left(9 a^{6} + 15 a^{5} + 12 a^{4} + 8 a^{3} + 17 a^{2} + 16 a + 18\right)\cdot 37^{2} + \left(19 a^{6} + 13 a^{5} + 34 a^{4} + 21 a^{3} + 7 a^{2} + 27 a + 4\right)\cdot 37^{3} + \left(24 a^{6} + 11 a^{5} + 36 a^{3} + 11 a^{2} + 30 a + 36\right)\cdot 37^{4} + \left(27 a^{6} + 32 a^{5} + 13 a^{4} + 30 a^{3} + 29 a^{2} + 28 a + 17\right)\cdot 37^{5} + \left(33 a^{6} + 29 a^{5} + 19 a^{4} + 18 a^{3} + a^{2} + 26 a + 17\right)\cdot 37^{6} + \left(29 a^{6} + 23 a^{5} + 6 a^{4} + 9 a^{3} + 26 a^{2} + 31 a + 31\right)\cdot 37^{7} + \left(16 a^{6} + 28 a^{5} + 25 a^{4} + 24 a^{3} + 11 a^{2} + 12 a + 26\right)\cdot 37^{8} +O(37^{9})$$ 31*a^6 + 28*a^5 + 27*a^4 + 17*a^3 + 3*a^2 + 3*a + 1 + (11*a^6 + 9*a^5 + 7*a^4 + 15*a^3 + 7*a^2 + 23*a + 34)*37 + (9*a^6 + 15*a^5 + 12*a^4 + 8*a^3 + 17*a^2 + 16*a + 18)*37^2 + (19*a^6 + 13*a^5 + 34*a^4 + 21*a^3 + 7*a^2 + 27*a + 4)*37^3 + (24*a^6 + 11*a^5 + 36*a^3 + 11*a^2 + 30*a + 36)*37^4 + (27*a^6 + 32*a^5 + 13*a^4 + 30*a^3 + 29*a^2 + 28*a + 17)*37^5 + (33*a^6 + 29*a^5 + 19*a^4 + 18*a^3 + a^2 + 26*a + 17)*37^6 + (29*a^6 + 23*a^5 + 6*a^4 + 9*a^3 + 26*a^2 + 31*a + 31)*37^7 + (16*a^6 + 28*a^5 + 25*a^4 + 24*a^3 + 11*a^2 + 12*a + 26)*37^8+O(37^9)

## Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

 Cycle notation $(1,9,11,6,12,4,7,13,10,3,5,2,8,14)$ $(1,13)(2,12)(3,11)(4,8)(5,6)(7,14)(9,10)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 14 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,13)(2,12)(3,11)(4,8)(5,6)(7,14)(9,10)$ $-1$ $1$ $7$ $(1,11,12,7,10,5,8)(2,14,9,6,4,13,3)$ $\zeta_{7}^{5}$ $1$ $7$ $(1,12,10,8,11,7,5)(2,9,4,3,14,6,13)$ $\zeta_{7}^{3}$ $1$ $7$ $(1,7,8,12,5,11,10)(2,6,3,9,13,14,4)$ $\zeta_{7}$ $1$ $7$ $(1,10,11,5,12,8,7)(2,4,14,13,9,3,6)$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ $1$ $7$ $(1,5,7,11,8,10,12)(2,13,6,14,3,4,9)$ $\zeta_{7}^{4}$ $1$ $7$ $(1,8,5,10,7,12,11)(2,3,13,4,6,9,14)$ $\zeta_{7}^{2}$ $1$ $14$ $(1,9,11,6,12,4,7,13,10,3,5,2,8,14)$ $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ $1$ $14$ $(1,6,7,3,8,9,12,13,5,14,11,4,10,2)$ $-\zeta_{7}^{4}$ $1$ $14$ $(1,4,5,9,7,2,11,13,8,6,10,14,12,3)$ $-\zeta_{7}^{2}$ $1$ $14$ $(1,3,12,14,10,6,8,13,11,2,7,9,5,4)$ $-\zeta_{7}^{5}$ $1$ $14$ $(1,2,10,4,11,14,5,13,12,9,8,3,7,6)$ $-\zeta_{7}^{3}$ $1$ $14$ $(1,14,8,2,5,3,10,13,7,4,12,6,11,9)$ $-\zeta_{7}$

The blue line marks the conjugacy class containing complex conjugation.