# Properties

 Label 1.75.10t1.a.a Dimension $1$ Group $C_{10}$ Conductor $75$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_{10}$ Conductor: $$75$$$$\medspace = 3 \cdot 5^{2}$$ Artin field: Galois closure of 10.0.37078857421875.1 Galois orbit size: $4$ Smallest permutation container: $C_{10}$ Parity: odd Dirichlet character: $$\chi_{75}(11,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{10} + 10x^{8} - 10x^{7} + 90x^{6} - 49x^{5} + 125x^{4} + 70x^{3} + 95x^{2} + 10x + 1$$ x^10 + 10*x^8 - 10*x^7 + 90*x^6 - 49*x^5 + 125*x^4 + 70*x^3 + 95*x^2 + 10*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{5} + 4x + 11$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.