# Properties

 Label 1.116.14t1.b.a Dimension $1$ Group $C_{14}$ Conductor $116$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{14}$ Conductor: $$116$$$$\medspace = 2^{2} \cdot 29$$ Artin field: 14.0.5796901408038404767744.1 Galois orbit size: $6$ Smallest permutation container: $C_{14}$ Parity: odd Dirichlet character: $$\chi_{116}(7,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{14} + 25 x^{12} + 214 x^{10} + 767 x^{8} + 1194 x^{6} + 686 x^{4} + 53 x^{2} + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 8.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.