# Properties

 Label 1.99.15t1.a.f Dimension $1$ Group $C_{15}$ Conductor $99$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{15}$ Conductor: $$99$$$$\medspace = 3^{2} \cdot 11$$ Artin field: 15.15.10943023107606534329121.1 Galois orbit size: $8$ Smallest permutation container: $C_{15}$ Parity: even Dirichlet character: $$\chi_{99}(31,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{5} + x + 14$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

 Cycle notation $(1,3,4,12,13)(2,6,9,7,11)(5,15,10,14,8)$ $(1,2,14)(3,6,8)(4,9,5)(7,15,12)(10,13,11)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 15 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,2,14)(3,6,8)(4,9,5)(7,15,12)(10,13,11)$ $-\zeta_{15}^{5} - 1$ $1$ $3$ $(1,14,2)(3,8,6)(4,5,9)(7,12,15)(10,11,13)$ $\zeta_{15}^{5}$ $1$ $5$ $(1,3,4,12,13)(2,6,9,7,11)(5,15,10,14,8)$ $\zeta_{15}^{6}$ $1$ $5$ $(1,4,13,3,12)(2,9,11,6,7)(5,10,8,15,14)$ $-\zeta_{15}^{7} - \zeta_{15}^{2}$ $1$ $5$ $(1,12,3,13,4)(2,7,6,11,9)(5,14,15,8,10)$ $\zeta_{15}^{3}$ $1$ $5$ $(1,13,12,4,3)(2,11,7,9,6)(5,8,14,10,15)$ $\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ $1$ $15$ $(1,6,5,12,11,14,3,9,15,13,2,8,4,7,10)$ $\zeta_{15}$ $1$ $15$ $(1,5,11,3,15,2,4,10,6,12,14,9,13,8,7)$ $\zeta_{15}^{2}$ $1$ $15$ $(1,11,15,4,6,14,13,7,5,3,2,10,12,9,8)$ $\zeta_{15}^{4}$ $1$ $15$ $(1,9,10,3,7,14,4,11,8,12,2,5,13,6,15)$ $\zeta_{15}^{7}$ $1$ $15$ $(1,15,6,13,5,2,12,8,11,4,14,7,3,10,9)$ $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ $1$ $15$ $(1,8,9,12,10,2,3,5,7,13,14,6,4,15,11)$ $-\zeta_{15}^{6} - \zeta_{15}$ $1$ $15$ $(1,7,8,13,9,14,12,6,10,4,2,15,3,11,5)$ $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ $1$ $15$ $(1,10,7,4,8,2,13,15,9,3,14,11,12,5,6)$ $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$

The blue line marks the conjugacy class containing complex conjugation.