Properties

Label 1.143.6t1.a.b
Dimension $1$
Group $C_6$
Conductor $143$
Root number not computed
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(143\)\(\medspace = 11 \cdot 13 \)
Artin field: Galois closure of 6.0.494190983.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{143}(10,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 34x^{4} - 35x^{3} + 162x^{2} + 36x + 467 \) Copy content 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 $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 26 a + 8 + \left(18 a + 12\right)\cdot 31 + \left(12 a + 15\right)\cdot 31^{2} + \left(27 a + 9\right)\cdot 31^{3} + \left(23 a + 19\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 30 a + 12 + \left(4 a + 4\right)\cdot 31 + \left(15 a + 15\right)\cdot 31^{2} + \left(26 a + 7\right)\cdot 31^{3} + \left(19 a + 23\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( a + 10 + \left(26 a + 15\right)\cdot 31 + \left(15 a + 9\right)\cdot 31^{2} + \left(4 a + 14\right)\cdot 31^{3} + \left(11 a + 5\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 a + 29 + \left(12 a + 23\right)\cdot 31 + \left(18 a + 21\right)\cdot 31^{2} + \left(3 a + 20\right)\cdot 31^{3} + \left(7 a + 8\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 a + 24 + \left(28 a + 25\right)\cdot 31 + 28\cdot 31^{2} + \left(8 a + 12\right)\cdot 31^{3} + \left(16 a + 21\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 22 a + 11 + \left(2 a + 11\right)\cdot 31 + \left(30 a + 2\right)\cdot 31^{2} + \left(22 a + 28\right)\cdot 31^{3} + \left(14 a + 14\right)\cdot 31^{4} +O(31^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.