Properties

Label 2.405.6t5.a.a
Dimension $2$
Group $S_3\times C_3$
Conductor $405$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(405\)\(\medspace = 3^{4} \cdot 5 \)
Artin stem field: 6.0.2460375.2
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.45.6t1.b.a
Projective image: $S_3$
Projective stem field: 3.1.135.1

Defining polynomial

$f(x)$$=$\(x^{6} + 6 x^{4} - x^{3} + 9 x^{2} - 3 x + 4\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \(x^{2} + 33 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 14 a + 35 + \left(22 a + 13\right)\cdot 37 + \left(21 a + 1\right)\cdot 37^{2} + \left(32 a + 13\right)\cdot 37^{3} + \left(31 a + 10\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 23 a + 17 + \left(14 a + 15\right)\cdot 37 + \left(15 a + 28\right)\cdot 37^{2} + \left(4 a + 10\right)\cdot 37^{3} + \left(5 a + 31\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 a + 33 + 7\cdot 37 + \left(2 a + 25\right)\cdot 37^{2} + \left(17 a + 34\right)\cdot 37^{3} + \left(26 a + 4\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 33 a + 24 + \left(21 a + 13\right)\cdot 37 + \left(19 a + 20\right)\cdot 37^{2} + \left(15 a + 28\right)\cdot 37^{3} + 5 a\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 19 a + 31 + \left(36 a + 28\right)\cdot 37 + \left(34 a + 32\right)\cdot 37^{2} + \left(19 a + 26\right)\cdot 37^{3} + \left(10 a + 19\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 8 + \left(15 a + 31\right)\cdot 37 + \left(17 a + 2\right)\cdot 37^{2} + \left(21 a + 34\right)\cdot 37^{3} + \left(31 a + 6\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.