Properties

Label 1.91.6t1.g.a
Dimension $1$
Group $C_6$
Conductor $91$
Root number not computed
Indicator $0$

Related objects

Learn more about

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(91\)\(\medspace = 7 \cdot 13 \)
Artin field: 6.0.9796423.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{91}(55,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} - 3 x^{4} + 7 x^{3} + 25 x^{2} - 55 x + 79\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \(x^{2} + 29 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 20 a + 23 + \left(25 a + 24\right)\cdot 31 + \left(17 a + 25\right)\cdot 31^{2} + \left(8 a + 5\right)\cdot 31^{3} + \left(27 a + 18\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 20 a + 21 + \left(25 a + 25\right)\cdot 31 + \left(17 a + 19\right)\cdot 31^{2} + \left(8 a + 16\right)\cdot 31^{3} + \left(27 a + 26\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 20 a + 5 + \left(25 a + 11\right)\cdot 31 + \left(17 a + 1\right)\cdot 31^{2} + \left(8 a + 25\right)\cdot 31^{3} + \left(27 a + 25\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 11 a + 14 + \left(5 a + 11\right)\cdot 31 + \left(13 a + 11\right)\cdot 31^{2} + \left(22 a + 24\right)\cdot 31^{3} + \left(3 a + 9\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 11 a + 1 + \left(5 a + 25\right)\cdot 31 + \left(13 a + 4\right)\cdot 31^{2} + \left(22 a + 5\right)\cdot 31^{3} + \left(3 a + 2\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 11 a + 30 + \left(5 a + 25\right)\cdot 31 + \left(13 a + 29\right)\cdot 31^{2} + \left(22 a + 15\right)\cdot 31^{3} + \left(3 a + 10\right)\cdot 31^{4} +O(31^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.