Properties

Label 2.280.6t5.a.b
Dimension $2$
Group $S_3\times C_3$
Conductor $280$
Root number not computed
Indicator $0$

Related objects

Learn more

Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Artin stem field: 6.0.3136000.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.280.6t1.d.b
Projective image: $S_3$
Projective stem field: 3.1.1960.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + 12 x^{2} - 20 x + 11\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 20 a + 20 + \left(2 a + 7\right)\cdot 29 + \left(3 a + 10\right)\cdot 29^{2} + \left(3 a + 22\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 14 a + 8 + \left(4 a + 6\right)\cdot 29 + \left(16 a + 28\right)\cdot 29^{2} + \left(10 a + 14\right)\cdot 29^{3} + \left(a + 9\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 24 a + 2 + \left(21 a + 15\right)\cdot 29 + \left(9 a + 19\right)\cdot 29^{2} + \left(15 a + 20\right)\cdot 29^{3} + \left(20 a + 7\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 5 a + 6 + \left(7 a + 13\right)\cdot 29 + \left(19 a + 17\right)\cdot 29^{2} + 13 a\cdot 29^{3} + \left(8 a + 8\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 9 a + 4 + \left(26 a + 1\right)\cdot 29 + \left(25 a + 23\right)\cdot 29^{2} + \left(25 a + 5\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 15 a + 20 + \left(24 a + 14\right)\cdot 29 + \left(12 a + 17\right)\cdot 29^{2} + \left(18 a + 22\right)\cdot 29^{3} + \left(27 a + 5\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$3$$2$$(1,6)(2,4)(3,5)$$0$
$1$$3$$(1,2,3)(4,5,6)$$-2 \zeta_{3} - 2$
$1$$3$$(1,3,2)(4,6,5)$$2 \zeta_{3}$
$2$$3$$(1,3,2)(4,5,6)$$-1$
$2$$3$$(4,6,5)$$\zeta_{3} + 1$
$2$$3$$(4,5,6)$$-\zeta_{3}$
$3$$6$$(1,5,2,6,3,4)$$0$
$3$$6$$(1,4,3,6,2,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.