# Properties

 Label 1.25.10t1.a.a Dimension $1$ Group $C_{10}$ Conductor $25$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $$25$$$$\medspace = 5^{2}$$ Artin field: Galois closure of $$\Q(\zeta_{25})^+$$ Galois orbit size: $4$ Smallest permutation container: $C_{10}$ Parity: even Dirichlet character: $$\chi_{25}(4,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{10} - 10x^{8} + 35x^{6} - x^{5} - 50x^{4} + 5x^{3} + 25x^{2} - 5x - 1$$ x^10 - 10*x^8 + 35*x^6 - x^5 - 50*x^4 + 5*x^3 + 25*x^2 - 5*x - 1 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{5} + 10x^{2} + 9$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.