# Properties

 Label 1.11.10t1.a.a Dimension $1$ Group $C_{10}$ Conductor $11$ Root number not computed Indicator $0$

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## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $$11$$ Artin number field: Galois closure of $$\Q(\zeta_{11})$$ Galois orbit size: $4$ Smallest permutation container: $C_{10}$ Parity: odd Dirichlet character: $$\chi_{11}(7,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{5} + 7 x + 28$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.