# Properties

 Label 1.49.7t1.a.b Dimension $1$ Group $C_7$ Conductor $49$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_7$ Conductor: $$49$$$$\medspace = 7^{2}$$ Artin field: Galois closure of 7.7.13841287201.1 Galois orbit size: $6$ Smallest permutation container: $C_7$ Parity: even Dirichlet character: $$\chi_{49}(36,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{7} - 21x^{5} - 21x^{4} + 91x^{3} + 112x^{2} - 84x - 97$$ x^7 - 21*x^5 - 21*x^4 + 91*x^3 + 112*x^2 - 84*x - 97 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{7} + 4x + 9$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.