# Properties

 Label 1.76.6t1.a.a Dimension $1$ Group $C_6$ Conductor $76$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$76$$$$\medspace = 2^{2} \cdot 19$$ Artin field: 6.0.8340544.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{76}(7,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 13 x^{4} + 50 x^{2} + 49$$  .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7 x + 2$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.