Properties

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

Related objects

Learn more

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(76\)\(\medspace = 2^{2} \cdot 19 \)
Artin field: 6.6.158470336.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{76}(27,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 19 x^{4} + 38 x^{2} - 19\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.