Properties

Label 2.648.6t5.b.b
Dimension $2$
Group $S_3\times C_3$
Conductor $648$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(648\)\(\medspace = 2^{3} \cdot 3^{4}\)
Artin stem field: 6.0.10077696.3
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.72.6t1.c.b
Projective image: $S_3$
Projective stem field: 3.1.216.1

Defining polynomial

$f(x)$$=$\(x^{6} + 6 x^{4} - 2 x^{3} + 9 x^{2} - 6 x + 7\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 2 + \left(2 a + 16\right)\cdot 17 + \left(7 a + 15\right)\cdot 17^{2} + \left(2 a + 4\right)\cdot 17^{3} + \left(2 a + 15\right)\cdot 17^{4} + \left(6 a + 8\right)\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 7 + \left(14 a + 13\right)\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + 14 a\cdot 17^{3} + \left(14 a + 15\right)\cdot 17^{4} + \left(10 a + 12\right)\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 a + 16 + \left(15 a + 3\right)\cdot 17 + \left(16 a + 14\right)\cdot 17^{2} + \left(7 a + 4\right)\cdot 17^{3} + 9\cdot 17^{4} + \left(9 a + 14\right)\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 13 + \left(a + 15\right)\cdot 17 + \left(7 a + 14\right)\cdot 17^{2} + \left(10 a + 3\right)\cdot 17^{3} + 2 a\cdot 17^{4} + \left(15 a + 15\right)\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 14 + \left(a + 4\right)\cdot 17 + 15\cdot 17^{2} + \left(9 a + 12\right)\cdot 17^{3} + \left(16 a + 1\right)\cdot 17^{4} + \left(7 a + 6\right)\cdot 17^{5} +O(17^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 14 a + 16 + \left(15 a + 13\right)\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + \left(6 a + 7\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} + \left(a + 10\right)\cdot 17^{5} +O(17^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.